A341039 -id:A341039 - cp's OEIS frontend

A341038 a(n) = Sum_{i+j<=m+1} d_i * d_j, where d_1 < ... < d_m are the divisors of n.

1, 5, 7, 17, 11, 39, 15, 49, 34, 59, 23, 144, 27, 79, 86, 129, 35, 198, 39, 219, 114, 119, 47, 436, 86, 139, 142, 287, 59, 523, 63, 321, 170, 179, 190, 760, 75, 199, 198, 676, 83, 690, 87, 423, 453, 239, 95, 1184, 162, 474, 254, 491, 107, 846, 278, 896, 282, 299, 119, 2061, 123, 319, 613, 769

Offset: 1

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

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

If p < q are primes, a(p*q) = 1 + 2*p + 2*q + p^2 + 4*p*q.

			The divisors of 6 are 1,2,3,6, so a(6) = 1*(1+2+3+6)+2*(1+2+3)+3*(1+2)+6*1 = 39.

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

Cf. A341039

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:
map(f, [$1..100]);

a(n) = my(d=divisors(n)); sum(k=1, #d, d[k]*sum(i=1, #d-k+1, d[i])); \\ Michel Marcus, Feb 04 2021

A341129 Numbers k such that A341117(k) is divisible by k and k is not a prime or power of a prime.

Original entry on oeis.org

42, 54, 66, 78, 102, 114, 135, 138, 147, 156, 162, 174, 186, 192, 222, 228, 246, 250, 258, 282, 318, 354, 366, 372, 402, 426, 438, 444, 474, 498, 507, 516, 534, 582, 606, 618, 642, 654, 678, 686, 732, 762, 786, 804, 822, 834, 845, 876, 894, 906, 942, 948, 978, 1002, 1029, 1038, 1074, 1083, 1086

Offset: 1

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

nonn

If k is a prime or power of a prime, A341117(k) is divisible by k.

Contains no semiprimes.

			a(3) = 66 is a term because 66 = 2*3*11 is not a prime or power of a prime and A341117(66) = 20262 = 66*307.

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

Cf. A341039, A341117.

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:
select(t -> not isprime(t) and nops(numtheory:-factorset(t))>1 and f(t) mod t = 0, [$2..10000]);

f(n) = my(d=divisors(n)); sum(k=1, #d, d[k]*sum(i=#d-k+1, #d, d[i])); \\ A341117
isok(m) = !(isprimepower(m) || (m==1)) && !(f(m) % m); \\ Michel Marcus, Feb 05 2021

cp's OEIS Frontend

A341038 a(n) = Sum_{i+j<=m+1} d_i * d_j, where d_1 < ... < d_m are the divisors of n.

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