A279048 -id:A279048 - cp's OEIS frontend

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

Antti Karttunen, Aug 08 2020

sign

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.

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

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

Cf. also A279048, A336835.

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 *)

A336915(n) = if(!bitand(n,n-1), -1, for(i=0,oo,my(n2 = n+n); if(sigma(n) >= n2, return(i)); n = n2));

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

A336916 One more than the number of iterations of x -> 2x needed before the result is nondeficient, when starting from x=n; a(n) = 0, when n is a power of 2.

Original entry on oeis.org

0, 0, 2, 0, 3, 1, 3, 0, 2, 2, 4, 1, 4, 2, 2, 0, 5, 1, 5, 1, 2, 3, 5, 1, 3, 3, 2, 1, 5, 1, 5, 0, 2, 4, 2, 1, 6, 4, 2, 1, 6, 1, 6, 2, 2, 4, 6, 1, 3, 2, 2, 2, 6, 1, 3, 1, 2, 4, 6, 1, 6, 4, 2, 0, 3, 1, 7, 3, 2, 1, 7, 1, 7, 5, 2, 3, 3, 1, 7, 1, 2, 5, 7, 1, 3, 5, 2, 1, 7, 1, 3, 3, 2, 5, 3, 1, 7, 2, 2, 1, 7, 1, 7, 1, 2

Offset: 1

Antti Karttunen, Aug 08 2020

nonn

See comments in A336915.

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

One more than A336915.

Cf. also A279048, A336835.

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

A336916(n) = if(!bitand(n,n-1), 0, for(i=1,oo,my(n2 = n+n); if(sigma(n) >= n2, return(i)); n = n2));

cp's OEIS Frontend

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.

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