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A336915 a(n) is the exponent of the least power of 2 that when multiplied by n, makes the product nondeficient, or -1 if n itself is a power of 2.

%I A336915 #19 Apr 01 2024 02:39:26
%S A336915 -1,-1,1,-1,2,0,2,-1,1,1,3,0,3,1,1,-1,4,0,4,0,1,2,4,0,2,2,1,0,4,0,4,
%T A336915 -1,1,3,1,0,5,3,1,0,5,0,5,1,1,3,5,0,2,1,1,1,5,0,2,0,1,3,5,0,5,3,1,-1,
%U A336915 2,0,6,2,1,0,6,0,6,4,1,2,2,0,6,0,1,4,6,0,2,4,1,0,6,0,2,2,1,4,2,0,6,1,1,0,6,0,6,0,1
%N A336915 a(n) is the exponent of the least power of 2 that when multiplied by n, makes the product nondeficient, or -1 if n itself is a power of 2.
%C A336915 Number of iterations of x -> 2x needed before the result is nondeficient (sigma(x) >= 2x), when starting from x=n, or -1 if a nondeficient number would never be reached (when n is a power of 2).
%C A336915 If neither x and y are powers of 2, and gcd(x,y) = 1, then a(x*y) <= min(a(x),a(y)). Compare to a similar comment in A336835.
%H A336915 Antti Karttunen, <a href="/A336915/b336915.txt">Table of n, a(n) for n = 1..65537</a>
%F A336915 For odd primes p, a(p) = A000523(p).
%t A336915 a[n_] := Module[{e = IntegerExponent[n, 2], s}, If[n == 2^e, -1, s = DivisorSigma[-1, n/2^e]; Max[Ceiling[Log2[s/(s - 1)]] - e - 1, 0]]]; Array[a, 100] (* _Amiram Eldar_, Apr 01 2024 *)
%o A336915 (PARI) A336915(n) = if(!bitand(n,n-1), -1, for(i=0,oo,my(n2 = n+n); if(sigma(n) >= n2, return(i)); n = n2));
%Y A336915 Cf. A000523, A005940, A336834, A336916 (same sequence + 1).
%Y A336915 Cf. also A279048, A336835.
%K A336915 sign
%O A336915 1,5
%A A336915 _Antti Karttunen_, Aug 08 2020