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A309978
a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.
3
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.
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
KEYWORD
nonn
AUTHOR
Peter Kagey, Aug 28 2019
STATUS
approved