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

1,2

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).

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A027750, A341038.

Sequence in context: A254541 A137658 A275277 * A255428 A367876 A216443

Adjacent sequences: A341114 A341115 A341116 * A341118 A341119 A341120

Keyword

nonn

Author

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

Status

approved