A341117 a(n) = Sum_{i+j>=m+1} d_i * d_j, where d_1 < ... < d_m are the divisors of n.

1, 8, 15, 44, 35, 129, 63, 208, 162, 305, 143, 712, 195, 553, 550, 912, 323, 1431, 399, 1665, 994, 1265, 575, 3356, 950, 1729, 1566, 3017, 899, 4901, 1023, 3840, 2266, 2873, 2254, 7845, 1443, 3553, 3094, 7744, 1763, 8862, 1935, 6897, 5901, 5129, 2303, 14672, 3234, 8475, 5134, 9425, 2915, 13986

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 05 2021

Keywords

• nonn

Comments

If p is prime, a(p^k) = (p^(2*k+2)-(2+k)*p^(k+1)+(k+1)*p^k)/(p - 1)^2.

If p < q are primes, a(p*q) = q*(p^2*q+2*p^2+2*p*q+4*p+q).

Examples

			The divisors of 6 are 1,2,3,6, so a(6) = 6*(1+2+3+6)+3*(2+3+6)+2*(3+6)+1*6 = 129.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A027750, A341038.

Programs

Maple

f:= proc(n) local D, S,i;
  D:= sort(convert(numtheory:-divisors(n),list),`>`);
  S:= ListTools:-PartialSums(D);
  add(D[i]*S[-i],i=1..nops(D))
end proc:
map(f, [$1..100]);

Mathematica

Array[Sum[#1[[k]]*Sum[#1[[j]], {j, #2 - k + 1, #2}], {k, #2}] & @@ {Divisors[#], DivisorSigma[0, #]} &, 54] (* Michael De Vlieger, Feb 05 2021 *)

PARI

a(n) = my(d=divisors(n)); sum(k=1, #d, d[k]*sum(i=#d-k+1, #d, d[i])); \\ Michel Marcus, Feb 05 2021