A277521 -id:A277521 - 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

A336723 a(n) = lcm(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).

Original entry on oeis.org

1, 6, 12, 168, 30, 36, 56, 960, 351, 900, 132, 12096, 182, 1176, 1800, 158720, 306, 75816, 380, 168000, 14112, 4356, 552, 1658880, 11625, 14196, 29160, 65856, 870, 810000, 992, 2064384, 17424, 31212, 58800, 917070336, 1406, 21660, 85176, 23040000, 1722, 6223392

Offset: 1

Jaroslav Krizek, Aug 01 2020

nonn

a(n) = pod(n) for numbers n: 1, 6, 30, 66, 84, 102, 120, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, ...

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

Cf. A009278 (lcm(tau(n), sigma(n))), A324528 (lcm(tau(n), pod(n))), A324529 (lcm(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = pod(k) and simultaneously A336722(k) = tau(k)).

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

a[n_] := LCM @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 50] (* Amiram Eldar, Aug 01 2020 *)

a(n) = my(d=divisors(n)); lcm([#d, vecsum(d), vecprod(d)]); \\ Michel Marcus, Aug 12 2020

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

A338395 Numbers m such that lcm(tau(m), sigma(m), pod(m)) = pod(m).

Original entry on oeis.org

1, 6, 30, 66, 84, 102, 120, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1080, 1092, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2040, 2130, 2226, 2280, 2310, 2346, 2370, 2604, 2670, 2730, 2760

Offset: 1

Jaroslav Krizek, Oct 23 2020

nonn

Numbers m such that A336723(m)= lcm(A000005(m), A000203(m), A007955(m)) = A007955(m).
Numbers m such that both values tau(m) and sigma(m) divide pod(m).
Numbers m such that all values m, tau(m) and sigma(m) divide pod(m); i.e. lcm(m, tau(m), sigma(m), pod(m)) = pod(m).
Supersequence of A277521.

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..12916 (a(n) < 10^7)
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..12916

Cf. A000005 (tau), A000203 (sigma), A007955 (pod).

Cf. A277521, A336723.

[m: m in [1..10^5] | LCM([#Divisors(m), &+Divisors(m), &*Divisors(m)]) eq &*Divisors(m)]

Select[Range[3000], LCM @@ {(d = DivisorSigma[0, #]), DivisorSigma[1, #], (pod = #^(d/2))} == pod &] (* Amiram Eldar, Oct 24 2020 *)

isok(m) = my(d=divisors(m), prd=vecprod(d)); lcm([#d, vecsum(d), prd]) == prd; \\ Michel Marcus, Oct 24 2020

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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A336723 a(n) = lcm(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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A338395 Numbers m such that lcm(tau(m), sigma(m), pod(m)) = pod(m).

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