A334782 -id:A334782 - cp's OEIS frontend

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

Jaroslav Krizek, May 10 2020

			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.

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

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))).

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

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

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

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

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

A334795 a(n) = Product_{d|n} lcm(d, tau(d)) where tau(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 6, 24, 10, 144, 14, 192, 54, 400, 22, 20736, 26, 784, 3600, 15360, 34, 23328, 38, 288000, 7056, 1936, 46, 3981312, 750, 2704, 5832, 790272, 58, 207360000, 62, 1474560, 17424, 4624, 19600, 120932352, 74, 5776, 24336, 92160000, 82, 796594176, 86, 3066624

Offset: 1

Jaroslav Krizek, May 12 2020

From Robert Israel, Jun 25 2020: (Start)
If p is an odd prime, a(p) = 2*p.
If p is a prime > 3, a(p^2) = 6*p^3.
If p and q are distinct odd primes, a(p*q) = 16*p^2*q^2. (End)

			a(6) = lcm(1, tau(1)) * lcm(2, tau(2)) * lcm(3, tau(3)) * lcm(6, tau(6)) = lcm(1, 1) * lcm(2, 2) * lcm(3, 2) * lcm(6, 4) = 1 * 2 * 6 * 12 = 144.

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

Cf. A334782 (Sum_{d|n} lcm(d, tau(d))), A334664 (Product_{d|n} gcd(d, tau(d))).

Cf. A000005 (tau(n)), A009230 (lcm(n, tau(n))).

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

g:= d -> ilcm(d, numtheory:-tau(d)):
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Jun 25 2020

a[n_] := Product[LCM[d, DivisorSigma[0, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], numdiv(d[k]))); \\ Michel Marcus, May 12 2020

a(p) = 2p for p = odd primes (A065091).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula