A334793 a(n) = Sum_{d|n} lcm(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

1, 3, 7, 27, 11, 45, 15, 91, 34, 113, 23, 1797, 27, 213, 917, 5211, 35, 5904, 39, 24137, 1785, 509, 47, 333637, 386, 705, 2950, 66093, 59, 811055, 63, 103515, 4385, 1193, 4925, 10085352, 75, 1485, 6117, 2584201, 83, 3113715, 87, 256085, 183194, 2165, 95

Offset

1,2

Links

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

Formula

a(p) = 2p + 1 for p = odd primes (A065091).

Example

a(6) = lcm(tau(1), pod(1)) + lcm(tau(2), pod(2)) + lcm(tau(3), pod(3)) + lcm(tau(6), pod(6)) = lcm(1, 1) + lcm(2, 2) + lcm(2, 3) + lcm(4, 36) = 1 + 2 + 6 + 36 = 45.

Maple

pod:= proc(n) option remember; convert(numtheory:-divisors(n), `*`) end proc:
f:= proc(n) local d; add(ilcm(numtheory:-tau(d), pod(d)), d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Jan 02 2025

Mathematica

a[n_] := DivisorSum[n, LCM[(d = DivisorSigma[0, #]), #^(d/2)] &]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)

Prog

(Magma) [&+[LCM(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]];
(PARI) a(n) = sumdiv(n, d, lcm(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 12 2020

Crossrefs

Cf. A334662 (Sum_{d|n} gcd(tau(d), pod(d))), A334784 (Sum_{d|n} lcm(tau(d), sigma(d))).

Cf. A000005 (tau(n)), A007955 (pod(n)), A324528 (lcm(tau(n), pod(n))).

Sequence in context: A126472 A019059 A305446 * A175198 A272530 A225038

Adjacent sequences: A334790 A334791 A334792 * A334794 A334795 A334796

Keyword

nonn,look

Author

Jaroslav Krizek, May 12 2020

Status

approved