A369894 -id:A369894 - cp's OEIS frontend

A369779 a(n) = n * Sum_{p|n, p prime} phi(n/p) / p.

0, 1, 1, 2, 1, 8, 1, 8, 6, 22, 1, 20, 1, 44, 26, 32, 1, 66, 1, 48, 48, 112, 1, 80, 20, 158, 54, 92, 1, 172, 1, 128, 116, 274, 62, 156, 1, 344, 162, 192, 1, 348, 1, 228, 174, 508, 1, 320, 42, 540, 278, 320, 1, 594, 130, 368, 348, 814, 1, 448, 1, 932, 306, 512, 176

Offset: 1

Wesley Ivan Hurt, Jan 31 2024

Dirichlet convolution of A010051(n) and A002618(n). - Wesley Ivan Hurt, Jul 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A002618, A010051, A057660, A117494, A347104, A369687.

Cf. also A369894, A369907, A369909.

Table[n*DivisorSum[n, EulerPhi[n/#]/# &, PrimeQ[#] &], {n, 100}]

A369779(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (eulerphi(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 23 2025

From Wesley Ivan Hurt, Jul 10 2025: (Start)
a(n) = Sum_{d|n} A010051(d) * A002618(n/d).
a(p^k) = ceiling(p^(2k-2)-p^(2k-3)) for p prime and k>=1. (End)

A369909 a(n) = n * Sum_{p|n, p prime} Omega(n/p) / p.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 8, 3, 7, 0, 20, 0, 9, 8, 24, 0, 30, 0, 28, 10, 13, 0, 60, 5, 15, 18, 36, 0, 62, 0, 64, 14, 19, 12, 90, 0, 21, 16, 84, 0, 82, 0, 52, 48, 25, 0, 160, 7, 70, 20, 60, 0, 135, 16, 108, 22, 31, 0, 186, 0, 33, 60, 160, 18, 122, 0, 76, 26, 118, 0, 240, 0

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001222 (Omega), A369741, A369908.

Cf. also A369894.

Table[n*DivisorSum[n, PrimeOmega[n/#]/# &, PrimeQ[#] &], {n, 100}]

A369909(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (bigomega(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 22 2025

a(p^k) = (k-1)*p^(k-1), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

A369907 a(n) = n * Sum_{p|n, p prime} omega(n/p) / p.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 4, 3, 7, 0, 16, 0, 9, 8, 8, 0, 21, 0, 24, 10, 13, 0, 32, 5, 15, 9, 32, 0, 62, 0, 16, 14, 19, 12, 60, 0, 21, 16, 48, 0, 82, 0, 48, 39, 25, 0, 64, 7, 45, 20, 56, 0, 63, 16, 64, 22, 31, 0, 154, 0, 33, 51, 32, 18, 122, 0, 72, 26, 118, 0, 120, 0, 39

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001221 (omega), A345354, A345355, A369744.

Cf. also A369894, A369909, A369779.

Table[n*DivisorSum[n, PrimeNu[n/#]/# &, PrimeQ[#] &], {n, 100}]

A369907(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (omega(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 23 2025

a(p^k) = p^(k-1), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

cp's OEIS Frontend

A369779 a(n) = n * Sum_{p|n, p prime} phi(n/p) / p.

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A369909 a(n) = n * Sum_{p|n, p prime} Omega(n/p) / p.

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A369907 a(n) = n * Sum_{p|n, p prime} omega(n/p) / p.

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Crossrefs

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Mathematica

PARI

Formula