A345312 a(n) = Sum_{p|n} (n/p)^gcd(p,n/p).

0, 1, 1, 4, 1, 5, 1, 16, 27, 7, 1, 40, 1, 9, 8, 64, 1, 225, 1, 104, 10, 13, 1, 152, 3125, 15, 729, 200, 1, 31, 1, 256, 14, 19, 12, 2052, 1, 21, 16, 408, 1, 41, 1, 488, 3384, 25, 1, 592, 823543, 100025, 20, 680, 1, 5859, 16, 792, 22, 31, 1, 932, 1, 33, 9270, 1024, 18, 61, 1

Offset

1,4

Comments

a(p) = Sum_{p|p} (p/p)^gcd(p,p/p) = 1^1 = 1 for primes p.

Links

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

Example

a(10) = Sum_{p|10} (10/p)^gcd(p,10/p) = 5^gcd(2,5) + 2^gcd(5,2) = 5^1 + 2^1 = 7.

Mathematica

Table[Sum[(n/k)^GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

Prog

(PARI) A345312(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, (n/f[i, 1])^gcd(n/f[i, 1], f[i, 1]))); \\ Antti Karttunen, Jan 22 2025

Crossrefs

Sequence in context: A028271 A168066 A029666 * A269593 A194512 A131230

Adjacent sequences: A345309 A345310 A345311 * A345313 A345314 A345315

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 13 2021

Status

approved