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

%I A181662 #35 Dec 14 2021 22:52:44
%S A181662 3,14,68,152,304,608,1984,3968,12032,24064,48128,96256,192512,385024,
%T A181662 770048,1540096,3080192,6160384,12320768,24641536,49283072,98566144,
%U A181662 197132288,394264576,788529152,1577058304,3154116608,6308233216,12616466432,25232932864,50465865728,100931731456
%N A181662 a(n) is the smallest positive integral multiple of 2^n not in the range of the Euler phi function.
%C A181662 From _Jianing Song_, Dec 14 2021: (Start)
%C A181662 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.
%C A181662 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).
%C A181662 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.
%C A181662 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)
%D A181662 David Harden, Posting to Sequence Fans Mailing List, Sep 19 2010.
%H A181662 Jianing Song, <a href="/A181662/a181662.txt">List of odd composites < 271129 such that the smallest n such that k * 2^n is a totient is greater than 100</a>.
%F A181662 a(n) = A058887(n)*2^n.
%Y A181662 Cf. A005277, A007617, A058887, A040076, A057192.
%K A181662 nonn
%O A181662 0,1
%A A181662 _N. J. A. Sloane_, Nov 18 2010
%E A181662 Escape clause removed by _Jianing Song_, Dec 14 2021