cp's OEIS Frontend

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).

%I A334793 #16 Jan 02 2025 20:04:07
%S A334793 1,3,7,27,11,45,15,91,34,113,23,1797,27,213,917,5211,35,5904,39,24137,
%T A334793 1785,509,47,333637,386,705,2950,66093,59,811055,63,103515,4385,1193,
%U A334793 4925,10085352,75,1485,6117,2584201,83,3113715,87,256085,183194,2165,95
%N 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).
%H A334793 Robert Israel, <a href="/A334793/b334793.txt">Table of n, a(n) for n = 1..10000</a>
%F A334793 a(p) = 2p + 1 for p = odd primes (A065091).
%e A334793 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.
%p A334793 pod:= proc(n) option remember; convert(numtheory:-divisors(n),`*`) end proc:
%p A334793 f:= proc(n) local d; add(ilcm(numtheory:-tau(d), pod(d)),d=numtheory:-divisors(n)) end proc:
%p A334793 map(f, [$1..100]); # _Robert Israel_, Jan 02 2025
%t A334793 a[n_] := DivisorSum[n, LCM[(d = DivisorSigma[0, #]), #^(d/2)] &]; Array[a, 100] (* _Amiram Eldar_, May 12 2020 *)
%o A334793 (Magma) [&+[LCM(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]];
%o A334793 (PARI) a(n) = sumdiv(n, d, lcm(numdiv(d), vecprod(divisors(d)))); \\ _Michel Marcus_, May 12 2020
%Y A334793 Cf. A334662 (Sum_{d|n} gcd(tau(d), pod(d))), A334784 (Sum_{d|n} lcm(tau(d), sigma(d))).
%Y A334793 Cf. A000005 (tau(n)), A007955 (pod(n)), A324528 (lcm(tau(n), pod(n))).
%K A334793 nonn,look
%O A334793 1,2
%A A334793 _Jaroslav Krizek_, May 12 2020