A397733 a(n) = Sum_{k=1..n} gcd(k^5,n).

1, 3, 5, 10, 9, 15, 13, 36, 33, 27, 21, 50, 25, 39, 45, 136, 33, 99, 37, 90, 65, 63, 45, 180, 145, 75, 261, 130, 57, 135, 61, 528, 105, 99, 117, 330, 73, 111, 125, 324, 81, 195, 85, 210, 297, 135, 93, 680, 385, 435, 165, 250, 105, 783, 189, 468, 185, 171, 117, 450, 121, 183

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = p^e + Sum_{k=0..e-1} (p^(e-k) - p^(e-k-1)) * p^min(5*k,e).

Equivalently, a(p^e) = p^e * ( 1 + ((p-1)/p) * Sum_{k=1..e} p^floor(4*k/5) ).

Let s and t be positive integers, and let a_{s,t}(n) = Sum_{k=1..n} gcd(k^s,n^t).

a_{s,t}(p^e) = p^min(s*e,t*e) + Sum_{k=0..e-1} (p^(e-k) - p^(e-k-1)) * p^min(s*k,t*e).

a_{s,t}(n) is multiplicative if and only if t <= s.

For t <= s, a_{s,t}(p^e) = p^e * ( 1 + ((p-1)/p) * Sum_{k=1..t*e} p^floor(((s-1)*k)/s) ).

Prog

(PARI) a(n) = sum(k=1, n, gcd(k^5, n));

Crossrefs

Cf. A018804, A078430, A372882, A393808.

Sequence in context: A078430 A372882 A393808 * A345892 A373210 A342424

Adjacent sequences: A397730 A397731 A397732 * A397735 A397736 A397737

Keyword

nonn,mult

Author

Seiichi Manyama, Jul 08 2026

Status

approved