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

%I A332411 #15 Feb 12 2020 08:18:31
%S A332411 0,1,3,1,25,7,343,1,9,101,14641,13,371293,2745,240,1,24137569,19,
%T A332411 893871739,401,9282,234257,78310985281,25,625,11881377,27,21953,
%U A332411 14507145975869,931,819628286980801,1,1185954,1544804417,44100,37,177917621779460413,114415582593,90224238,1601
%N A332411 If n = Product (p_j^k_j) then a(n) = Sum (n^(pi(p_j) - 1)), where pi = A000720.
%H A332411 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A332411 a(n) = [x^n] Sum_{k>=1} n^(k - 1) * x^prime(k) / (1 - x^prime(k)).
%e A332411 a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 21^1 + 21^3 = 9282;
%e A332411 9282 in base 21 (reverse order of digits with leading zero) = 0101.
%e A332411                                                                | |
%e A332411                                                                2 4
%p A332411 a:= n-> add(n^(numtheory[pi](i[1])-1), i=ifactors(n)[2]):
%p A332411 seq(a(n), n=1..42);  # _Alois P. Heinz_, Feb 11 2020
%t A332411 a[n_] := Plus @@ (n^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 40}]
%t A332411 Table[SeriesCoefficient[Sum[n^(k - 1) x^Prime[k]/(1 - x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 1, 40}]
%o A332411 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, n^(primepi(f[k,1])-1)); \\ _Michel Marcus_, Feb 11 2020
%Y A332411 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.
%K A332411 nonn
%O A332411 1,3
%A A332411 _Ilya Gutkovskiy_, Feb 11 2020