A347104 -id:A347104 - 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)

A369687 a(n) = Sum_{p|n, p prime} p^phi(n/p).

Original entry on oeis.org

0, 2, 3, 2, 5, 7, 7, 4, 9, 21, 11, 13, 13, 71, 106, 16, 17, 73, 19, 41, 778, 1035, 23, 97, 625, 4109, 729, 113, 29, 362, 31, 256, 59170, 65553, 18026, 145, 37, 262163, 531610, 881, 41, 4874, 43, 1145, 22186, 4194327, 47, 6817, 117649, 1049201, 43047010, 4265, 53, 262873, 9780266, 6497

Offset: 1

Wesley Ivan Hurt, Jan 28 2024

Cf. A000010 (phi), A347104, A369779, A369782.

Table[DivisorSum[n, #^EulerPhi[n/#] &, PrimeQ[#] &], {n, 60}]

from sympy import totient, primefactors
def A369687(n): return sum(p**totient(n//p) for p in primefactors(n)) # Chai Wah Wu, Jan 28 2024

a(p^k) = p^((p-1)*p^(k-2)+floor(1/k)/p) for p prime and k>=1. - Wesley Ivan Hurt, Jul 16 2025

A369782 a(n) = Sum_{p|n, p prime} n^phi(n/p).

Original entry on oeis.org

0, 2, 3, 4, 5, 42, 7, 64, 81, 10010, 11, 288, 13, 7529550, 50850, 65536, 17, 34012548, 19, 160400, 85766562, 26559922791446, 23, 663552, 390625, 95428956661682202, 387420489, 481891088, 29, 656100810900, 31, 1099511627776, 1531578985265538, 3189059870763703892770850

Offset: 1

Wesley Ivan Hurt, Jan 31 2024

Cf. A000010, A345893, A347104, A369687, A369779.

Table[DivisorSum[n, n^EulerPhi[n/#] &, PrimeQ[#] &], {n, 40}]

a(p^k) = p^(k*(p-1)*p^(k-2)+floor(1/k)/p) for p prime and k>=1. - Wesley Ivan Hurt, Jul 16 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369687 a(n) = Sum_{p|n, p prime} p^phi(n/p).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Formula

A369782 a(n) = Sum_{p|n, p prime} n^phi(n/p).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula