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A332411
If n = Product (p_j^k_j) then a(n) = Sum (n^(pi(p_j) - 1)), where pi = A000720.
0
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
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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2 4
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 11 2020
STATUS
approved