A181662 a(n) is the smallest positive integral multiple of 2^n not in the range of the Euler phi function.

3, 14, 68, 152, 304, 608, 1984, 3968, 12032, 24064, 48128, 96256, 192512, 385024, 770048, 1540096, 3080192, 6160384, 12320768, 24641536, 49283072, 98566144, 197132288, 394264576, 788529152, 1577058304, 3154116608, 6308233216, 12616466432, 25232932864, 50465865728, 100931731456

Offset: 0

Author

N. J. A. Sloane, Nov 18 2010

Keywords

• nonn

Comments

From Jianing Song, Dec 14 2021: (Start)

Let a(n) = 2^n * k, then k must be odd, otherwise a(n)/2 is a totient number, which implies that a(n) is a totient.

Note that 271129 * 2^m is a nontotient for all m (see A058887), so k <= 271129. In fact, let p be smallest prime such that 2^e*p + 1 is composite for all 0 <= e <= n, then k <= p (since 2^n*p is a nontotient).

Actually, k is equal to p. To verify this, it suffices to show that k cannot be an odd composite number < 271129; that is to say, if 2^n * k is a nontotient for an odd composite number < 271129, then there exists k' < k such that 2^n * k' is a nontotient.

The case k < 383 can be easily checked. Let k be an odd composite number in the range (383, 271129), k * 2^n is a nontotient implies n < 2554 unless k = 98431 or 248959 (see the a-file below), then 383 * 2^n is a nontotient (the least n such that 383 * 2^n + 1 is prime is n = 6393). For k = 98431 or 248959, k * 2^n is a nontotient implies n < 7062, then 2897 * 2^n is a nontotient (the least n such that 2897 * 2^n + 1 is prime is n = 9715. (End)

References

• David Harden, Posting to Sequence Fans Mailing List, Sep 19 2010.

Links

• Jianing Song, List of odd composites < 271129 such that the smallest n such that k * 2^n is a totient is greater than 100.

Crossrefs

Cf. A005277, A007617, A058887, A040076, A057192.

Formula

a(n) = A058887(n)*2^n.

Extensions

Escape clause removed by Jianing Song, Dec 14 2021