REAVLEY, Circuit Judge:

In this consolidated appeal, Erik Wayerski, Raymond Roy, John Mosman, and Stepan Bondarenko appeal their convictions and sentences for engaging in multiple child-pornography offenses. All four co-defendants were convicted of engaging in a child exploitation enterprise, in violation of 18 U.S.C. § 2252A(g); conspiring to advertise, transport, receive, and possess child pornography and to obstruct justice, in violation of 18 U.S.C. §§ 371, 1512(k), 2251(d)(1) and (e), and 2252A(a)(1) and (b)(1); and advertising the exchange of child pornography, in violation of 18 U.S.C. § 2251(d)(1). In addition, Wayerski and Roy were convicted of transporting and shipping child pornography, in violation of 18 U.S.C. § 2252A(a)(1), and receipt of child pornography, in violation of 18 U.S.C. § 2252A(a)(2). Finally, Wayerski was also convicted of obstruction of justice, in violation of 18 U.S.C. § 1512(c)(2).

The defendants raise five issues on appeal in their individual briefs and by adopting each other's arguments. Their primary issue is that the statute of conviction for engaging in a child exploitation enterprise, 18 U.S.C. § 2252A(g), is unconstitutionally vague. We hold that § 2252A(g) is not impermissibly vague, and we affirm the convictions of all defendants. However, we also agree with the Government's concession of error that the defendants' convictions for both engaging in a child exploitation enterprise and for a conspiracy premised on the same acts was impermissible. We reject all other claims of error, and we remand to the district court for the appropriate modification of the judgments.

I.

All of the defendants argue that their convictions under 18 U.S.C. § 2252A(g) for engaging in a child exploitation enterprise should be vacated because § 2252A(g) is unconstitutionally vague under the Fifth Amendment. They assert that § 2252A(g) fails to provide fair notice of what conduct it proscribes, does not provide law enforcement with sufficient standards, and leads to absurd results not intended by Congress. We disagree.

We review whether a criminal statute is unconstitutionally vague de novo. United States v. Duran.1 Where, as in this case, a vagueness challenge does not involve the First Amendment, the analysis must be as applied to the facts of the case.2 "Vagueness" is an outgrowth of the Fifth Amendment's Due Process Clause. United States v. Williams.3 It encompasses notions of fair warning such that people of common intellect may understand a statute's prohibitions and need not guess at its meaning. United States v. Hunt.4 A criminal statute will violate due process if it "fails to provide people of ordinary intelligence a reasonable opportunity to understand what conduct it prohibits" or "it authorizes or even encourages arbitrary and discriminatory enforcement." Hill v. Colorado.5 There is a strong presumption that statutes passed by Congress are valid. United States v. Nat'l Dairy Prods. Corp.6

The first step in a vagueness inquiry is to examine the plain language of the statute.7 "The touchstone of the inquiry is the meaning of the statute in light of common understanding and practice."8 When the plain text of the statute sets forth clearly perceived boundaries, our inquiry is ended.9

Under § 2252A(g), a person who engages in a "child exploitation enterprise" is subject to imprisonment "for any term of years not less than 20 or for life."10 The statute provides that

The defendants argue that the statute is vague because it is unclear how a "series of felony violations constituting three or more separate incidents" may occur. We do not find the statute impermissibly vague, however, especially in light of the facts of the instant case.12 We agree with the Government that a person of ordinary intelligence would, under a fair reading of the statute, understand that an offense is committed when (1) he commits any of several enumerated predicate offenses, (2) the predicate offenses comprise a series of three felony violations on three or more separate instances, (3) the offense involves more than one victim, and (4) he commits the predicate offenses in concert with three or more other persons.

In the instant case, the defendants participated in a sophisticated group of approximately 45 individuals who advertised and exchanged over the Internet thousands of images and videos of child pornography involving numerous minor children. An investigation by police lasting approximately two years revealed that the defendants used complex encryption technology and procedures to facilitate and conceal their activity. The defendants communicated with each other by using innocent newsgroup websites, such as for cooking or music, to post secret messages that would lead group members to the location of child pornography on other sites, as well as instructions on how to decrypt the material. The group limited its membership to known traders of child pornography who had been recommended by another member. Members were instructed to use aliases and to follow a written set of security measures and standard operating procedures to further the exchange of the child pornography. During the course of the group's existence over 400,000 images and videos which had traveled in interstate commerce via the Internet were advertised, transported, and/or received by its members. Group members also made specific requests for and purchased the production of new illicit material.

Nothing about § 2252A(g) is vague when applied in the context of the defendants' actions. Section 2252A(g) defines the predicate offenses that must be committed. The defendants' activity here satisfied the predicate offenses for § 2252A(g) because it violated at a minimum the prohibitions of 18 U.S.C. § 2251(d)(1), concerning the advertisement of child pornography, and 18 U.S.C. § 2252A(a)(1) and (2), concerning the transportation and receipt of child pornography, which are Chapter 110 offenses covered in the child exploitation enterprise statute. The offenses involved much more than three separate instances and more than one victim, and they occurred in concert with more than three people. Indeed, the defendants do not suggest that their own conduct falls outside the reach of § 2252A(g). We therefore reject their vagueness challenge, as "`[o]ne to whose conduct a statute clearly applies may not successfully challenge it [facially] for vagueness.'" Bama Tomato Co. v. U.S. Dep't of Agriculture.13

The defendants argue that the statute is vague because the word "series" is either undefined or is ambiguous as to whether it differs from the requirement that there be multiple incidents. We are not persuaded by the defendants' hypertechnical reading of the statute, however.14 We agree with the Government that the statute clarifies that "series" means "three or more separate incidents" of the enumerated predicate felony violations.

Defendants also present several hypotheticals purporting to show the statute's vagueness. For example, they argue that, contrary to Congress' intent, the statute may be unwittingly violated in a matter of minutes by a defendant who receives and transports three or more images of child pornography involving more than one child to three or more other persons. Even assuming the defendants could present their facial challenge, however, "speculation about possible vagueness in hypothetical situations not before the Court will not support a facial attack on a statute when it is surely valid in the vast majority of its intended applications."15 We think "`it is clear what the [statute] as a whole prohibits.'"16 It forbids the commission of specified child pornography offenses that occur as a series of three or more separate instances, involving two or more victims, and three or more persons acting in concert with the defendant. Section 2252A(g)'s plain language covered the defendants' conduct and satisfied the due process requirement for fair notice.17

We therefore hold that § 2252A(g) is not unconstitutionally vague as applied to the defendants.

II.

Roy, Mosman, and Bondarenko argue that Count 1 of the indictment, which alleged that the defendants engaged in a child exploitation enterprise, failed to allege sufficiently a violation of § 2252A(g). They assert that Count 1 violated the Fifth and Sixth Amendments because it failed to identify both the statutes and the actions constituting the predicate offenses for a child exploitation enterprise. Consequently, they claim that they were not provided with adequate notice of the Government's case and there was no assurance that the grand jury found probable cause.

We review the sufficiency of an indictment de novo. United States v. Bobo.18 In "analyzing challenges to the sufficiency of an indictment, courts give the indictment a common sense construction, and its validity is to be determined by practical, not technical, considerations." United States v. Poirier.19

An indictment is sufficient when it "(1) presents the essential elements of the charged offense, (2) notifies the accused of the charges to be defended against, and (3) enables the accused to rely upon a judgment under the indictment as a bar against double jeopardy for any subsequent prosecution for the same offense." United States v. Woodruff.20 These requirements satisfy the Sixth Amendment's guarantee of notice to the accused of the nature and the cause of the accusation, and the Fifth Amendment's assurance that a grand jury will return an indictment only when it finds probable cause for all elements of the crime. United States v. Fern.21 "If an indictment specifically refers to the statute on which the charge was based, the reference to the statutory language adequately informs the defendant of the charge."22 "Moreover, the constitutional standard is fulfilled by an indictment that tracks the wording of the statute, as long as the language sets forth the essential elements of the crime." United States v. Ndiaye.23 When the facts alleged in the indictment permit an inference that the grand jury found probable cause, the indictment satisfies the Fifth Amendment.24

In the instant case, Count 1 of the indictment charged the defendants as follows:

Under a practical interpretation of the indictment, Count 1 was plainly sufficient. It specifically referred to and tracked the language of the statute on which it was based, i.e. § 2252A(g), and provided notice to the defendants of the charges to be defended. It alleged that the defendants engaged in a child exploitation enterprise with specifically named co-defendants by advertising, transporting, and shipping child pornography in interstate commerce, as part of a series of three or more separate instances and involving more than one victim. Furthermore, it alleged the time and place of the predicate violations. The indictment was adequate to apprise the defendants of the charges and to plead double jeopardy in any future prosecution for the same offense. See, e.g., United States v. Bascaro.25

Because Count 1 of the indictment cited § 2252A(g), tracked § 2252A(g)'s language, and provided a general description of the facts and predicate offenses, Count 1 sufficiently alleged a violation of § 2252A(g).

III.

Next, all of the defendants argue that their convictions for both engaging in a child exploitation enterprise and conspiring to commit multiple child-pornography offenses violated the Fifth Amendment's Double Jeopardy Clause. The Government concedes, and we agree, that the defendants are correct.

The Double Jeopardy Clause of the Fifth Amendment protects "against multiple punishments for the same offense." United States v. Nyhuis.26 To determine whether a defendant has been punished twice for the same offense, we apply the test of Blockburger v. United States. See Rutledge v. United States.27 Under the Blockburger test, "[i]f the same act or transaction constitutes a violation of two distinct statutory provisions, the test to be applied to determine whether there are two offenses or only one, is whether each provision requires proof of a fact which the other does not."28 Accordingly, "two different statutes define the `same offense,' typically because one is a lesser included offense of the other."29

The defendants contend that their conspiracy convictions are subsumed within their convictions for a child exploitation enterprise because § 2252A(g)'s requirement that at least three individuals act "in concert" to form an "enterprise" is essentially a conspiracy, and as such, their conspiracy convictions are lesser included offenses. In support of their argument, defendants rely on Rutledge.

In Rutledge, the Supreme Court held that a drug conspiracy conviction, in violation of 21 U.S.C. § 846, was a lesser included offense of a continuing criminal enterprise conviction, in violation of 21 U.S.C. § 848.30 The Court reasoned that the requirement for a continuing criminal enterprise offense that the defendant have acted "in concert" with others required proof of an agreement that would also violate the conspiracy statute.31 Because the conspiracy offense did not require proof of any fact that was not also part of the continuing criminal enterprise offense, a straightforward application of the Blockburger test resulted in the conclusion that the conspiracy statute did not define a separate offense and was a lesser included offense of the more serious crime for a continuing criminal enterprise.32

Similarly here, § 2252A(g) requires proof that a defendant acted "in concert" with at least three other persons. This conduct is an element of the child exploitation enterprise offense that requires the same proof of an agreement that would also violate the conspiracy offense charged in Count 2 of the defendants' indictment.33 Because the defendants' conspiracy convictions did not require proof of facts different from the child exploitation enterprise offense's "in concert" requirement, we hold that the defendants' conspiracy convictions were lesser included offenses and violated the Double Jeopardy Clause. Accordingly, we remand to the district court with instructions to vacate the defendants' conspiracy convictions. As a consequence of our decision, we vacate the judgment insofar as the conspiracy count is concerned. In light of that action we also vacate the sentence and remand for resentencing, but we "commit the decisions as to whether to conduct a resentencing hearing and as to the scope of any such hearing to the discretion of the district court." See United States v. Gari.34

IV.

Roy and Mosman argue that the district court erred by applying a two-level sentence enhancement for obstruction of justice pursuant to U.S.S.G. § 3C1.1. However, we find no clear error.35

The Sentencing Guidelines provide for a two-level enhancement to the base offense level if

The presentence investigation report detailed numerous obstructive actions that the defendants took during the course of their child exploitation enterprise to avoid detection and thwart any law enforcement investigation. For example, as a group the defendants used continuously changing encryption keys to decipher and encrypt material posted to newsgroups; periodically moved from one newsgroup to another and changed their nicknames; used sophisticated computer file swapping techniques requiring special instructions to re-assemble files; and used special software programs that could lock down a computer or wipe it clean should law enforcement enter a member's home.

Roy and Mosman make no argument that these actions were not obstructive conduct. They argue that the obstruction enhancement was inapplicable because, even though they took numerous precautions to avoid detection, they were not under arrest and did not know that they were under investigation at the time they took these precautions. This argument is unavailing. The Guidelines' commentary specifically provides that "[o]bstructive conduct that occurred prior to the start of the investigation of the instant offense of conviction" may be covered by the Guideline.37 Moreover, we have previously recognized that "[t]here is no requirement that defendant's obstructive acts occur subsequent to the formal commencement of an investigation[.]" United States v. Garcia.38 The defendants' affirmative steps to prevent law enforcement from detecting their illicit activity and to impede any investigation show that they consciously acted with the purpose of obstructing justice. See United States v. Massey.39

We also reject Roy's and Mosman's contention that the district court erred by failing to make detailed findings setting forth the conduct that amounted to obstruction of justice. There is no need for a district court to make specific findings where, as in this case, it both adopts a presentence investigation report that contains specific findings and the defendant fails to request that the court make more specific findings. United States v. Smith.40

Because there is no requirement that a defendant be under arrest or know he is being investigated at the time he commits obstructive acts, and the defendants failed to request that the district court make more specific findings in addition to those contained in the presentence investigation report, the district court did not err in applying a two-level obstruction-of-justice enhancement.

V.

Finally, Wayerski and Roy argue that their sentences were unreasonable. Both defendants were sentenced within their respective Guidelines ranges. Wayerski was sentenced at the top of his range to 365 months in prison, and Roy was sentenced to 360 months, at the bottom of his range.

We review the reasonableness of a sentence under an abuse-of-discretion standard. Gall v. United States.41 A defendant challenging his sentence bears the burden of establishing that it is unreasonable. United States v. Talley.42 On appeal, the analysis of the reasonableness of a sentence is a two-step process. United States v. Pugh.43 First, we examine whether the district court committed any significant procedural error. Second, after it has been determined that a sentence is procedurally sound, we review the sentence's substantive reasonableness.44

When reviewing for procedural reasonableness, we ensure that the district court: (1) properly calculated the Guidelines range; (2) treated the Guidelines as advisory; (3) considered the 18 U.S.C. § 3553(a) factors; (4) did not select a sentence based on clearly erroneous facts; and (5) adequately explained the chosen sentence.45

Substantive reasonableness involves examining the totality of the circumstances and whether the sentence is supported by the sentencing factors outlined in § 3553(a). See United States v. Gonzalez.46 The § 3553(a) factors are:

There is an expectation of reasonableness when a district court imposes a sentence within the applicable Guidelines range.48

Neither Wayerski nor Roy can show that their sentences are unreasonable. In both cases, the district court properly calculated the Guideline range, treated the range as advisory, considered the statutory sentencing factors, and imposed a sentence within the Guideline range that is supported by the § 3353(a) factors.49

Wayerski argues that his sentence is unreasonable because the district court improperly assumed that he would pose a significant risk to children after he was released from prison, even though there was no support in the record for such an assumption and he had no prior criminal history. He relies in part on the opinion of psychologist Dr. James Larson, who conducted a court-ordered examination of Wayerski and concluded that the risk of Wayerski re-offending once he is out of prison was low. The district court acknowledged Dr. Larson's opinion but noted that other evidence in the record supported a conclusion that Wayerski could be a danger to the community. The evidence included surreptitious videotapes of neighborhood children that Wayerski filmed while making sexually explicit comments and expressing a desire to rape the children. In addition, the presentence investigation report noted that Wayerski had written, and posted on-line, diaries about an eight-year old child who lived next door to him in which he described various sexual acts he desired to commit upon the girl and her friends. Wayerski had also attempted to videotape the girl through her bedroom window while she was undressing. Dr. Larson apparently reached his conclusion about Wayerski without reviewing the diaries or videos Wayerski created. Furthermore, a search of Wayerski's home revealed a pair of child's underwear that matched a pair of panties the child's mother said were missing. In light of the record, the district court's conclusion that Wayerski posed a possible threat to the community was not clearly erroneous.

At sentencing, the district court specifically noted the above evidence and indicated that it had considered the § 3553(a) factors, particularly the need to protect the public and provide deterrence. The weight the district court gave to these factors was within its sound discretion.50 Wayerski fails to show that the district court abused its discretion and imposed an unreasonable sentence.51

Roy argues that his sentence is substantively unreasonable because the Sentencing Guidelines for child-pornography offenses are irrational and not empirically based, his criminal history category was misleading, and his conduct was not so serious at to warrant the sentence he received. We are unpersuaded. Roy fails to show how the totality of the circumstances or the sentencing factors of § 3353(a) render his sentence unreasonable. The district court sentenced Roy to the bottom of his Guideline range specifically because of Roy's efforts to cooperate with the Government. The court noted that it had considered the § 3353(a) factors and believed a greater sentence was not necessary, but that the sentence imposed served the sentencing goals of punishment and deterrence. It is evident that the district court carefully considered the § 3553(a) sentencing factors, and we can find no fault in the district court's reasoning.

Both Wayerski and Roy argue that their conduct of receiving, transmitting, and exchanging child pornography is insufficiently serious to warrant the harsh sentences they received. We agree with the Government, however, that the defendants understate the severity of their conduct and the harm caused by their offenses. We have previously recognized the harm caused by child pornography offenses. Those who receive and exchange child pornography create a demand that influences the production of the pornography and the attendant physical and emotional injury to children.52 "`The logic of deterrence suggests that the lighter the punishment for downloading and uploading child pornography, the greater the customer demand for it and so the more will be produced.'"53 It was therefore not unreasonable for the district court to consider the defendants' offenses to be serious and to conclude that significant sentences were warranted.

Because both Wayerski and Roy failed to demonstrate that the district court abused its discretion, both of their sentences were reasonable. We therefore affirm the district court's judgments as to the total sentence imposed.

Conclusion

Upon review of the record and consideration of the parties' arguments, we affirm except that the Count 2 convictions are vacated. The case is remanded for the district court for resentencing, as discussed above.

AFFIRMED IN PART, VACATED AND REMANDED IN PART.