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.

-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, -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, 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

Offset: 1

Author

Antti Karttunen, Aug 08 2020

Keywords

• sign

Comments

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).

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000523, A005940, A336834, A336916 (same sequence + 1).

Cf. also A279048, A336835.

Programs

Mathematica

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 *)

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));

Formula

For odd primes p, a(p) = A000523(p).