A334794
a(n) = Sum_{d|n} lcm(sigma(d), pod(d)) where sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).
Original entry on oeis.org
1, 7, 13, 63, 31, 55, 57, 1023, 364, 937, 133, 12207, 183, 1239, 1843, 32767, 307, 76222, 381, 168993, 14181, 4495, 553, 1672047, 3906, 14385, 29524, 23247, 871, 812785, 993, 2097151, 17569, 31525, 58887, 917158710, 1407, 22047, 85371, 23209953, 1723, 6238791
Offset: 1
a(6) = lcm(sigma(1), pod(1)) + lcm(sigma(2), pod(2)) + lcm(sigma(3), pod(3)) + lcm(sigma(6), pod(6)) = lcm(1, 1) + lcm(3, 2) + lcm(4, 3) + lcm(12, 36) = 1 + 6 + 12 + 36 = 55.
Cf.
A334663 (Sum_{d|n} gcd(sigma(d), pod(d))),
A334793 (Sum_{d|n} lcm(tau(d), pod(d))).
-
[&+[LCM(&+Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
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a[n_] := DivisorSum[n, LCM[DivisorSigma[1, #], #^(DivisorSigma[0, #]/2)] &]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)
-
a(n) = sumdiv(n, d, lcm(sigma(d), vecprod(divisors(d)))); \\ Michel Marcus, May 12 2020
A334807
a(n) = Product_{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).
Original entry on oeis.org
1, 2, 6, 48, 10, 432, 14, 3072, 162, 2000, 22, 17915904, 26, 5488, 54000, 15728640, 34, 68024448, 38, 1152000000, 148176, 21296, 46, 380420285792256, 3750, 35152, 472392, 8674025472, 58, 314928000000000, 62, 1546188226560, 574992, 78608, 686000
Offset: 1
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 = 432.
Cf.
A334793 (Sum_{d|n} lcm(tau(d), pod(d))),
A334730 (Product_{d|n} gcd(tau(d), pod(d))).
-
[&*[LCM(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]];
-
pod:= proc(n) option remember; convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; mul(ilcm(numtheory:-tau(d), pod(d)),d=numtheory:-divisors(n)) end proc:
map(f, [$1..50]); # Robert Israel, Jan 02 2025
-
a[n_] := Product[LCM[DivisorSigma[0, d], Times @@ Divisors[d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)
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a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(numdiv(d[k]), vecprod(divisors(d[k])))); \\ Michel Marcus, Jun 27 2020
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