A309978 a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.

0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1

Offset

1,4

Comments

Records occur at 1, 2, 4, 6, 30, ...

Does there exist n such that a(n) >= 5? Do there exist examples besides 30 and 130 such that a(n) = 4? If so in either case, n > A253913(10000) = 87469256.

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Formula

a(2n+1) = 1 for all n >= 1.

a(2n) >= 2 for all n >= 2.

Example

For n = 130 the a(130) = 4 positive integers with valid maps are
  129 via 129 + 129^0 = 130,
   65 via  65 +  65^1 = 130,
    5 via   5 +   5^3 = 130, and
    2 via   2 +   2^7 = 130.

Prog

(PARI) a(n) = {if (n==1, return (0)); my(d = divisors(n)); 1 + sumdiv(n, d, if ((d>1) && (d<n) && d^valuation(n-d, d) == n-d, 1)); } \\ Michel Marcus, Oct 16 2019

Crossrefs

Cf. A253913, A307074, A307092, A309997.

Sequence in context: A162696 A370265 A364447 * A108103 A111376 A241664

Adjacent sequences: A309975 A309976 A309977 * A309979 A309980 A309981

Keyword

nonn

Author

Peter Kagey, Aug 28 2019

Status

approved