A309997 -id:A309997 - cp's OEIS frontend

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

Peter Kagey, Aug 28 2019

nonn

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.

			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.

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

Cf. A253913, A307074, A307092, A309997.

a(n) = {if (n==1, return (0)); my(d = divisors(n)); 1 + sumdiv(n, d, if ((d>1) && (dMichel Marcus, Oct 16 2019

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

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

A328422 Number of paths from 2 to n via maps of the form x -> x + x^j, where j is a nonnegative integer.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 14, 14, 18, 18, 24, 24, 31, 31, 42, 42, 51, 51, 65, 65, 79, 79, 97, 97, 118, 118, 142, 142, 167, 167, 198, 198, 229, 229, 271, 271, 317, 317, 368, 368, 419, 419, 484, 484, 549, 549, 628, 628, 707, 707, 808, 808, 905, 905, 1023

Offset: 2

Peter Kagey, Oct 15 2019

nonn

This sequence is essentially the same as the number of paths from 1 to n. However, starting from 2 removes the ambiguity of how many maps there are from 1 to 2.

a(2n+1) = a(2n) for all n because x + x^j is odd if and only if x is even and j = 0.

			For n = 8 the a(8) = 6 paths are:
2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 with j = [0,0,0,0,0,0]
2 -> 3 -> 4 -> 8                with j = [0,0,1]
2 -> 3 -> 6 -> 7 -> 8           with j = [0,1,0,0]
2 -> 4 -> 5 -> 6 -> 7 -> 8      with j = [1,0,0,0,0]
2 -> 4 -> 8                     with j = [1,1]
2 -> 6 -> 7 -> 8                with j = [2,0,0]

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A307074, A307092, A309997, A309978.

a(2) = 1, a(n) = Sum_{k=1..A309978(n)} a(A328446(n,k)) for n > 2.

cp's OEIS Frontend

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

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