A332411 If n = Product (p_j^k_j) then a(n) = Sum (n^(pi(p_j) - 1)), where pi = A000720.

0, 1, 3, 1, 25, 7, 343, 1, 9, 101, 14641, 13, 371293, 2745, 240, 1, 24137569, 19, 893871739, 401, 9282, 234257, 78310985281, 25, 625, 11881377, 27, 21953, 14507145975869, 931, 819628286980801, 1, 1185954, 1544804417, 44100, 37, 177917621779460413, 114415582593, 90224238, 1601

Offset: 1

Author

Ilya Gutkovskiy, Feb 11 2020

Keywords

• nonn

Examples

			a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 21^1 + 21^3 = 9282;
9282 in base 21 (reverse order of digits with leading zero) = 0101.
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Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000079 (without a(0) gives the positions of 1's), A000244 (without a(0) gives the fixed points), A000720, A087207, A090883, A276379 (a(n) written in base n), A308814.

Programs

Maple

a:= n-> add(n^(numtheory[pi](i[1])-1), i=ifactors(n)[2]):
seq(a(n), n=1..42);  # Alois P. Heinz, Feb 11 2020

Mathematica

a[n_] := Plus @@ (n^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 40}]
Table[SeriesCoefficient[Sum[n^(k - 1) x^Prime[k]/(1 - x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 1, 40}]

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, n^(primepi(f[k,1])-1)); \\ Michel Marcus, Feb 11 2020

Formula

a(n) = [x^n] Sum_{k>=1} n^(k - 1) * x^prime(k) / (1 - x^prime(k)).