A334783 a(n) = Sum_{d|n} lcm(d, sigma(d)).

1, 7, 13, 35, 31, 31, 57, 155, 130, 127, 133, 143, 183, 231, 163, 651, 307, 382, 381, 575, 741, 535, 553, 383, 806, 735, 1210, 315, 871, 631, 993, 2667, 673, 1231, 1767, 3770, 1407, 1527, 2379, 1055, 1723, 1599, 1893, 1487, 1450, 2215, 2257, 2367, 2850, 5552

Offset: 1

Author

Jaroslav Krizek, May 10 2020

Keywords

• nonn
• look

Examples

			a(6) = lcm(1, sigma(1)) + lcm(2, sigma(2)) + lcm(3, sigma(3)) + lcm(6, sigma(6)) = lcm(1, 1) + lcm(2, 3) + lcm(3, 4) + lcm(6, 12) = 1 + 6 + 12 + 12 = 31.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, Plot of a(n)/n^2 for n=1..20000

Crossrefs

Cf. A334490 (Sum_{d|n} gcd(d, sigma(d))), A334782 (Sum_{d|n} lcm(d, tau(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009242 (lcm(n, sigma(n))).

Programs

Magma

[&+[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N do
  t:= ilcm(d,numtheory:-sigma(d));
  R:= [seq(i,i=d..N,d)];
  V[R]:= V[R] +~ t;
od:
convert(V,list); # Robert Israel, May 13 2020

Mathematica

a[n_] := DivisorSum[n, LCM[#, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)

PARI

a(n) = sumdiv(n, d, lcm(d, sigma(d))); \\ Michel Marcus, May 10 2020

Formula

a(p) = p^2 + p + 1 for p = primes (A000040).