A306682 -id:A306682 - cp's OEIS frontend

A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 1, 8, 1, 8, 1, 2, 1, 12, 1, 4, 1, 1, 1, 8, 1, 2, 1, 8, 1, 3, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 1, 3, 1, 1, 8, 1, 2, 1

Offset: 1

Jaroslav Krizek, Aug 01 2020

nonn

a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...

			a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).

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

a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* Amiram Eldar, Aug 01 2020 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ Antti Karttunen, Aug 10 2020

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

a(n) = gcd(A007955(n), A009205(n)). - Antti Karttunen, Aug 10 2020

Data section extended up to a(105) by Antti Karttunen, Aug 10 2020

A334663 a(n) = Sum_{d|n} gcd(sigma(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 2, 3, 2, 15, 2, 4, 3, 5, 2, 20, 2, 7, 6, 5, 2, 19, 2, 8, 4, 7, 2, 33, 3, 5, 4, 64, 2, 93, 2, 6, 6, 5, 4, 25, 2, 7, 4, 19, 2, 69, 2, 12, 10, 7, 2, 38, 3, 7, 12, 8, 2, 44, 4, 73, 4, 5, 2, 124, 2, 7, 6, 7, 4, 167, 2, 8, 6, 27, 2, 41, 2, 5, 8, 12, 4, 43, 2

Offset: 1

Jaroslav Krizek, May 07 2020

nonn

Inverse Möbius transform of A306682. - Antti Karttunen, May 09 2020

			a(6) = gcd(sigma(1), pod(1)) + gcd(sigma(2), pod(2)) + gcd(sigma(3), pod(3)) + gcd(sigma(6), pod(6)) = gcd(1, 1) + gcd(3, 2) + gcd(4, 3) + gcd(12, 36) = 1 + 1 + 1 + 12 = 15.

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

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).
Cf. A000203 (sigma(n)), A007955 (pod(n)), A306682 (gcd(sigma(n), pod(n))).
Cf. A334731 (product instead of sum).

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

a(n) = sumdiv(n, d, gcd(sigma(d), vecprod(divisors(d)))); \\ Michel Marcus, May 08 2020

a(p) = 2 for p = primes (A000040).

A334731 a(n) = Product_{d|n} gcd(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, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 48, 1, 4, 3, 1, 1, 36, 1, 4, 1, 4, 1, 576, 1, 2, 1, 224, 1, 5184, 1, 1, 3, 2, 1, 144, 1, 4, 1, 40, 1, 2304, 1, 16, 9, 4, 1, 2304, 1, 2, 9, 4, 1, 864, 1, 1792, 1, 2, 1, 995328, 1, 4, 1, 1, 1, 20736, 1, 4, 3, 128, 1, 5184, 1, 2

Offset: 1

Jaroslav Krizek, May 09 2020

nonn

			a(6) = gcd(sigma(1), pod(1)) * gcd(sigma(2), pod(2)) * gcd(sigma(3), pod(3)) * gcd(sigma(6), pod(6)) = gcd(1, 1) * gcd(3, 2) * gcd(4, 3) * gcd(12, 36) = 1 * 1 * 1 * 12 = 12.

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

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A306682 (gcd(sigma(n), pod(n))).

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

a[n_] := Product[GCD[DivisorSigma[1, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)

pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(sigma(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

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

A324526 Numbers m such that gcd(sigma(m), pod(m)) = tau(m) where tau(k) = the number of divisors of k (A000005), sigma(k) = the sum of the divisors of k (A000203) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 14, 22, 38, 46, 56, 62, 86, 94, 110, 118, 134, 142, 150, 158, 166, 184, 206, 214, 254, 262, 278, 286, 302, 326, 334, 342, 358, 374, 382, 398, 422, 430, 446, 454, 478, 486, 494, 502, 504, 526, 542, 566, 568, 612, 614, 622, 638, 646, 662, 670, 694, 718, 726

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

Numbers n such that A306682(n) = A000005(n).

			14 is a term because gcd(sigma(14), pod(14)) = gcd(24, 196) = 4 = tau(14).

Cf. A000005, A000203, A007955, A306682.

[n: n in [1..10^5] | GCD(SumOfDivisors(n), &*[d: d in Divisors(n)]) eq NumberOfDivisors(n)]

isok(n) = my(d=divisors(n)); gcd(vecsum(d), vecprod(d)) == #d; \\ Michel Marcus, Mar 05 2019

A324527 a(n) = the smallest number m such that gcd(sigma(m), pod(m)) = n where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 10, 15, 12, 95, 180, 91, 56, 51, 40, 473, 6, 117, 980, 135, 70, 1139, 90, 703, 290, 861, 26378, 3151, 54, 745, 468, 255, 2156, 5017, 26100, 775, 124, 1419, 2176, 4865, 96, 2701, 26714, 585, 190, 6683, 65268, 11051, 5632, 435, 144946, 13207, 42, 679, 5800

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

a(n) = the smallest number m such that A306682(m) = n.

			For n=2; a(2) = 10 because gcd(sigma(10), pod(10)) = gcd (18, 100) = 2 and 10 is the smallest.

Cf. A000005, A000203, A007955, A074391, A306682.

[Min([n: n in[1..10^5] | GCD(SumOfDivisors(n), &*[d: d in Divisors(n)]) eq k]): k in [1..45]]

f(n) = my(d=divisors(n)); gcd(vecsum(d), vecprod(d)); \\ A306682
a(n) = {my(k=1); while (f(k) != n, k++); k;} \\ Michel Marcus, Mar 05 2019

cp's OEIS Frontend

A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

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A334663 a(n) = Sum_{d|n} gcd(sigma(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

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A334731 a(n) = Product_{d|n} gcd(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).

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A324526 Numbers m such that gcd(sigma(m), pod(m)) = tau(m) where tau(k) = the number of divisors of k (A000005), sigma(k) = the sum of the divisors of k (A000203) and pod(n) = the product of divisors of k (A007955).

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A324527 a(n) = the smallest number m such that gcd(sigma(m), pod(m)) = n where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

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