A332653 -id:A332653 - cp's OEIS frontend

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

[&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^n.

a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} k^n.

A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 4, 15, 76, 785, 7836, 137263, 2130976, 47895489, 1010012140, 28531167071, 743044702104, 25239592216033, 797785008119932, 31147773583464735, 1157442765678719056, 51702516367896047777, 2185932446984222457444, 109912203092239643840239, 5255987282125826560192520

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031972, A056665, A057661, A226561, A228640, A332620, A332621, A332653, A332655.

[&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^k, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^k.
a(n) = n * A332653(n).

A365633 The sum of divisors of n that are terms of A072873.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

The number of these divisors is A365632(n) and the largest of them is A327939(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A072873, A327939, A332653, A365632.

f[p_, e_] := (p^(Floor[e/p] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(1+f[i,2] \ f[i,1])-1)/(f[i,1] - 1));}

Multiplicative with a(p^e) = (p^(floor(e/p)+1) - 1)/(p - 1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (A332653(p)/(p^(p-1)-1) - 1/(p*(p-1))) = 2.253624924813... .

cp's OEIS Frontend

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

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A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

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Author

Keywords

Crossrefs

Programs

Magma

Mathematica

Formula

A365633 The sum of divisors of n that are terms of A072873.

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