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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.

%I A309978 #71 Oct 16 2019 13:08:48
%S A309978 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,
%T A309978 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,
%U A309978 1,2,1,3,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1
%N A309978 a(n) is the number of positive integers k such that there exists a nonnegative integer m with k + k^m = n.
%C A309978 Records occur at 1, 2, 4, 6, 30, ...
%C A309978 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.
%H A309978 Peter Kagey, <a href="/A309978/b309978.txt">Table of n, a(n) for n = 1..10000</a>
%F A309978 a(2n+1)  = 1 for all n >= 1.
%F A309978 a(2n)   >= 2 for all n >= 2.
%e A309978 For n = 130 the a(130) = 4 positive integers with valid maps are
%e A309978   129 via 129 + 129^0 = 130,
%e A309978    65 via  65 +  65^1 = 130,
%e A309978     5 via   5 +   5^3 = 130, and
%e A309978     2 via   2 +   2^7 = 130.
%o A309978 (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
%Y A309978 Cf. A253913, A307074, A307092, A309997.
%K A309978 nonn
%O A309978 1,4
%A A309978 _Peter Kagey_, Aug 28 2019