A341038 -id:A341038 - cp's OEIS frontend

A341039 Numbers k such that k divides A341038(k).

1, 12, 18, 56, 117, 810, 992, 2091, 16256, 796797, 2488734, 22122192, 57636005, 67100672, 114243682

Offset: 1

J. M. Bergot and Robert Israel, Feb 03 2021

Contains no primes.

			a(3) = 18 is a term because A341038(18) = 198 is divisible by 18.

Includes A139256.

Cf. A341038.

f:= proc(n) local D,S,i;
  D:= sort(convert(numtheory:-divisors(n),list));
  S:= ListTools:-PartialSums(D);
  add(S[-i]*D[i],i=1..nops(D))
end proc:
select(t -> f(t) mod t = 0, [$1..20000]);

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

Original entry on oeis.org

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

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

nonn

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).

			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.

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

Cf. A027750, A341038.

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]);

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

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

cp's OEIS Frontend

A341039 Numbers k such that k divides A341038(k).

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