A341621 a(n) is the exponent of the least power of 2 that when multiplied by n makes the product abundant, or -1 if n itself is a power of 2.

-1, -1, 2, -1, 2, 1, 3, -1, 1, 1, 3, 0, 3, 2, 1, -1, 4, 0, 4, 0, 1, 2, 4, 0, 2, 2, 1, 1, 4, 0, 5, -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, 4, 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, Feb 19 2021

Keywords

• sign

Comments

Number of iterations of x -> 2x needed before the result is abundant (sigma(x) > 2x), when starting from x=n, or -1 if an abundant number would never be reached (when n is a power of 2).

Links

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

Crossrefs

Cf. A005101 (positions of zeros), A341622 (positions where this differs from A336915).

Programs

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2], s}, If[n == 2^e, -1, s = DivisorSigma[-1, n/2^e]; Max[Floor[Log2[s/(s - 1)]] - e, 0]]]; Array[a, 100] (* Amiram Eldar, Apr 01 2024 *)

PARI

A341621(n) = if(!bitand(n,n-1), -1, for(i=0,oo,my(n2 = n+n); if(sigma(n) > n2, return(i)); n = n2));