A238983 -id:A238983 - cp's OEIS frontend

A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

1, 5, 28, 257, 3126, 47450, 823544, 16777217, 387420490, 10009766650, 285311670612, 8916117756914, 302875106592254, 11112685048647250, 437893920912786408, 18446744073709551617, 827240261886336764178, 39346558169931834836690, 1978419655660313589123980

Offset: 1

José María Grau Ribas, Mar 07 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A034448, A238982, A238983.

a[n_, k_] := Sum[If[GCD[i, n] == i && GCD[i, n/i] == 1, i^k, 0], {i, n}]; Table[a[n, n], {n, 1, 24}]
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(n * Last[#]) &/@ FactorInteger[n]);  Array[a, 19] (* Amiram Eldar, Aug 10 2019 *)

a(n) = sumdiv(n, d, d^n*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 19 2014

For prime p, a(p) = p^p + 1; A125137 is a subsequence. - Michel Marcus, Nov 20 2015
a(n) = n^n+1 (A014566) if n is a prime power (A246655). - Michel Marcus, Nov 21 2015
a(n) = Sum_{d|n, gcd(d,n/d)=1} d^n. - Wesley Ivan Hurt, Apr 28 2023

A238982 Numbers n dividing the sum of n-th powers of unitary divisors of n.

Original entry on oeis.org

1, 10, 45, 50, 130, 250, 315, 410, 735, 1125, 1250, 1690, 2050, 2205, 2210, 2373, 2565, 2745, 3045, 3250, 3285, 3321, 3465, 3645, 4225, 5050, 5330, 6125, 6250, 6615, 6890, 7875, 8619, 8835, 9135, 9225, 9555, 9933, 10250

Offset: 1

José María Grau Ribas, Mar 07 2014

nonn

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A034448, A238981, A238983.

udivs:= proc(n) local t,F;
  F:= map(t -> t[1]^t[2], ifactors(n)[2]);
  map(convert, combinat:-powerset(F),`*`);
end proc:
filter:= proc(n) local t,U;
  convert(map(t -> (t &^ n) mod n, udivs(n)),`+`) mod n = 0
end proc:
select(filter, [$1..20000]); # Robert Israel, Dec 07 2022

AA[n_, k_] := AA[n, k] = Mod[Sum[If[GCD[i, n] == i && GCD[i, n/i] == 1, PowerMod[i, k, n], 0], {i, n}], n]; Select[Range[1000], Mod[AA[#, #], #] == 0 &]

cp's OEIS Frontend

A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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