A295576 -id:A295576 - cp's OEIS frontend

A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

0, 1, 1, 1, 5, 1, 14, 10, 21, 10, 55, 26, 91, 35, 70, 84, 204, 75, 285, 140, 210, 165, 506, 196, 525, 286, 549, 406, 1015, 340, 1240, 680, 880, 680, 1190, 654, 2109, 969, 1482, 1080, 2870, 966, 3311, 1650, 2010, 1771, 4324, 1544, 4214, 2050

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

n does not divide a(n) iff n = (2^k)*(q^m) with k > 0, m >= 0 and q odd prime such that q == 3 (mod 4) or n = (2^k)*(3^L)*Product_{q} q^(v_q) with k >= 0, L > 0, v_q >= 0 and all q odd primes such that q == 5 (mod 6). - René Gy, Oct 21 2018

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.
René Gy, The sum of product pairs of integers prime to n, Math StackExchange.

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.

Cf. A023022.

R:=proc(n,k) local x,t1,S;
t1:={}; S:=0;
for x from 1 to floor(n/2) do if gcd(x,n)=1 then t1:={op(t1),x^k}; S:=S+x^k; fi; od;
S; end;
s:=k->[seq(R(n,k),n=1..50)];
s(2);

f[n_] := Plus @@ (Select[ Range[n/2], GCD[#, n] == 1 &]^2); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^2); \\ Michel Marcus, Dec 10 2017

A295575 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^3.

Original entry on oeis.org

0, 1, 1, 1, 9, 1, 36, 28, 73, 28, 225, 126, 441, 153, 416, 496, 1296, 469, 2025, 1100, 1710, 1225, 4356, 1800, 4959, 2556, 5581, 4410, 11025, 3872, 14400, 8128, 11090, 8128, 15822, 8910, 29241, 13041, 21996, 16400, 44100, 15426, 53361, 27830, 33716, 29161, 76176, 27936, 77652, 37828

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

If p is an odd prime, a(p) = (n^2-1)^2/64. - Robert Israel, Dec 10 2017

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.

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.

f:= n -> add(t^3, t = select(t->igcd(t,n)=1, [$1..n/2])) :
map(f, [$1..100]); # Robert Israel, Dec 10 2017

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^3); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^3); \\ Michel Marcus, Dec 10 2017

cp's OEIS Frontend

A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

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