A369913 -id:A369913 - cp's OEIS frontend

A369912 a(n) = Sum_{p|n, p prime} p^sopf(n/p).

0, 1, 1, 4, 1, 17, 1, 4, 27, 57, 1, 41, 1, 177, 368, 4, 1, 251, 1, 153, 2530, 2169, 1, 41, 3125, 8361, 27, 561, 1, 5568, 1, 4, 178478, 131361, 94932, 275, 1, 524649, 1596520, 153, 1, 37514, 1, 8313, 6686, 8389137, 1, 41, 823543, 78157, 129145076, 32937, 1, 251

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Cf. A008472 (sopf), A369744, A369911, A369913.

a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, #^a[n/#] &, PrimeQ[#] &], {n, 80}]

a(p^k) = p^(p-p*floor(1/k)) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025

A369914 a(n) = n * Sum_{p|n, p prime} sopf(n/p) / p.

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 8, 9, 29, 0, 38, 0, 53, 34, 16, 0, 57, 0, 78, 58, 125, 0, 76, 25, 173, 27, 134, 0, 220, 0, 32, 130, 293, 74, 150, 0, 365, 178, 156, 0, 366, 0, 294, 147, 533, 0, 152, 49, 195, 298, 398, 0, 171, 146, 268, 370, 845, 0, 500, 0, 965, 237, 64, 194

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Dirichlet convolution of c(n) and n * sopf(n), where c = A010051. - Wesley Ivan Hurt, Jul 10 2025

Cf. A008472 (sopf), A010051, A369744, A369911, A369912, A369913.

a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[n*DivisorSum[n, a[n/#]/# &, PrimeQ[#] &], {n, 80}]

a(p^k) = p^k * (1 - floor(1/k)) for p prime and k>=1. - Wesley Ivan Hurt, Jul 09 2025

a(n) = n * Sum_{d|n} c(d) * sopf(n/d) / d, where c = A010051. - Wesley Ivan Hurt, Jul 10 2025

cp's OEIS Frontend

A369912 a(n) = Sum_{p|n, p prime} p^sopf(n/p).

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