A334490 -id:A334490 - cp's OEIS frontend

A334491 a(n) = Product_{d|n} gcd(d, sigma(d)).

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 24, 1, 2, 3, 1, 1, 18, 1, 4, 1, 2, 1, 288, 1, 2, 1, 56, 1, 216, 1, 1, 3, 2, 1, 72, 1, 2, 1, 40, 1, 72, 1, 8, 9, 2, 1, 1152, 1, 2, 3, 4, 1, 108, 1, 448, 1, 2, 1, 20736, 1, 2, 1, 1, 1, 216, 1, 4, 3, 8, 1, 2592, 1, 2, 3, 8, 1, 72

Offset: 1

Jaroslav Krizek, May 03 2020

nonn

			a(6) = gcd(1, sigma(1)) * gcd(2, sigma(2)) * gcd(3, sigma(3)) * gcd(6, sigma(6)) = gcd(1, 1) * gcd(2, 3) * gcd(3, 4) * gcd(6, 12) = 1 * 1 * 1 * 6 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A334490 (Sum_{d|n} gcd(d, sigma(d))), A007955 (pod(n) = Product_{d|n} gcd(d, pod(d))).

Cf. A000040, A000203 (sigma(n)), A009194.

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

a[n_] := Product[GCD[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 80] (* Amiram Eldar, May 03 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(d[k], sigma(d[k]))); \\ Michel Marcus, May 03-11 2020

a(p) = 1 for p = primes (A000040).

a(n) = Product_{d|n} A009194(d). - Antti Karttunen, May 09 2020

A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

Original entry on oeis.org

1, 2, 3, 3, 3, 8, 3, 4, 4, 6, 3, 11, 3, 8, 9, 5, 3, 12, 3, 13, 9, 8, 3, 16, 4, 6, 8, 11, 3, 24, 3, 8, 9, 6, 9, 16, 3, 8, 9, 16, 3, 26, 3, 15, 16, 8, 3, 19, 6, 10, 9, 9, 3, 24, 9, 20, 9, 6, 3, 45, 3, 8, 12, 9, 9, 26, 3, 13, 9, 24, 3, 24, 3, 6, 12, 11, 9, 24, 3

Offset: 1

Jaroslav Krizek, May 06 2020

nonn

Inverse Möbius transform of A009205. - Antti Karttunen, May 19 2020

			a(6) = gcd(tau(1), sigma(1)) + gcd(tau(2), sigma(2)) + gcd(tau(3), sigma(3)) + gcd(tau(6), sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(2, 4) + gcd(4, 12) = 1 + 1 + 2 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334490 (Sum_{d|n} gcd(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).
Cf. A334729 (with product, instead of sum).

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

a[n_] := DivisorSum[n, GCD[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 07 2020 *)

a(n) = sumdiv(n, d, gcd(numdiv(d), sigma(d))); \\ Michel Marcus, May 07 2020

a(p) = 3 for p = odd primes (A065091).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A334491 a(n) = Product_{d|n} gcd(d, sigma(d)).

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A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

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

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