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

1,3

Links

Table of n, a(n) for n=1..40.

Index entries for sequences computed from indices in prime factorization

Formula

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

Example

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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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}]

Prog

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

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.

Sequence in context: A309397 A193472 A259208 * A104033 A217472 A156654

Adjacent sequences: A332408 A332409 A332410 * A332412 A332413 A332414

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 11 2020

Status

approved