A295576 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^4.

0, 1, 1, 1, 17, 1, 98, 82, 273, 82, 979, 626, 2275, 707, 2674, 3108, 8772, 3027, 15333, 9044, 14994, 9669, 39974, 17668, 50085, 24310, 60597, 50470, 127687, 45604, 178312, 103496, 149908, 103496, 225302, 129750, 432345, 187017, 349830, 266088, 722666

Offset

1,5

Comments

If p is an odd prime, a(p) = p*(p^2-1)*(3*p^2-7)/480. - Robert Israel, Dec 10 2017

Links

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

John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.

Formula

From Chai Wah Wu, Apr 28 2026: (Start)

If n==0 (mod 4): a(n) = n*(3*n^3*A000010(n)+20*n^2*A023900(n)-8*A063453(n))/480

If n is odd: a(n) = n*(3*n^3*A000010(n)-10*n^2*A023900(n)+7*A063453(n))/480

If n!=2 and n==2 (mod 4): a(n) = n*(21*n^3*A000010(n)-280*n^2*A023900(n)+64*A063453(n))/3360 (End)

Maple

f:= n -> add(t^4, t = select(t->igcd(t, n)=1, [$1..n/2])):
map(f, [$1..100]); # Robert Israel, Dec 10 2017

Mathematica

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^4); Array[f, 41] (* Robert G. Wilson v, Dec 10 2017 *)

Prog

(PARI) a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^4); \\ Michel Marcus, Dec 10 2017
(Python)
from math import prod
from sympy import primefactors
def A295576(n):
    if n == 2: return 1
    m, ps = n&3, primefactors(n)
    s1, s3 = prod(1-p for p in ps), prod(1-p**3 for p in ps)
    t = n*(-s1 if len(ps)&1 else s1)//prod(ps)
    if not m:
        return (3*n**4*t+20*n**3*s1-8*n*s3)//480
    elif m==2:
        return n*(21*n**3*t - 280*n**2*s1 + 64*s3)//3360
    else:
        return n*(3*n**3*t - 10*n**2*s1 + 7*s3)//480 # Chai Wah Wu, Apr 28 2026

Crossrefs

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

Sequence in context: A102292 A264439 A279363 * A376019 A223519 A139804

Adjacent sequences: A295573 A295574 A295575 * A295577 A295578 A295579

Keyword

nonn

Author

N. J. A. Sloane, Dec 08 2017

Status

approved