A336722 -id:A336722 - cp's OEIS frontend

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).

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).

A337323 a(n) = gcd(n, 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, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1

Offset: 1

Jaroslav Krizek, Aug 23 2020

GCD(n, tau(n), sigma(n), pod(n)) = GCD(n, tau(n), sigma(n)). - David A. Corneth, Aug 24 2020

			a(6) = gcd(6, tau(6), sigma(6), pod(6)) = gcd(6, 4, 12, 36) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A337324 (least m such that gcd(m, tau(m), sigma(m), pod(m)) = n).
Cf. A336723 (lcm(tau(n), sigma(n), pod(n))) = (lcm(n, tau(n), sigma(n), pod(n))).
Cf. A000005, A000203, A007955.

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

f:= proc(n) uses numtheory; igcd(n, tau(n), sigma(n)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2020

a[n_] := GCD @@ {n, DivisorSigma[0, n], DivisorSigma[1, n]}; Array[a, 100] (* Amiram Eldar, Aug 24 2020 *)

a(n) = my(f=factor(n)); gcd([n, sigma(f), numdiv(f)]); \\ Michel Marcus, Apr 01 2021

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

a(n) = 1 for n = p^k, p prime, k >= 0 (A000961). - Bernard Schott, Apr 01 2021

A329929 a(n) = lcm(tau(n), sigma(n), pod(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).

Original entry on oeis.org

1, 6, 12, 168, 30, 9, 56, 960, 351, 450, 132, 6048, 182, 294, 1800, 158720, 306, 25272, 380, 84000, 14112, 1089, 552, 414720, 11625, 7098, 29160, 32928, 870, 101250, 992, 2064384, 17424, 15606, 58800, 917070336, 1406, 5415, 85176, 11520000, 1722, 777924, 1892

Offset: 1

Jaroslav Krizek, Aug 31 2020

nonn

a(n) is also lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).

			a(6) = lcm(tau(6), sigma(6), pod(6)) / gcd(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) / gcd(4, 12, 36) = 36 / 4 = 9.

Cf. A336722, A336723, A337323.
Cf. A334985 (lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, tau(n), sigma(n), pod(n))).
Cf. A000005, A000203, A007955.

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

a[n_] := LCM @@ (t = {(d = DivisorSigma[0, n]), n^(d/2), DivisorSigma[1, n]}) / GCD @@ t; Array[a, 50] (* Amiram Eldar, Aug 31 2020 *)

a(n) = my(f=factor(n), v=[numdiv(f), sigma(f), vecprod(divisors(f))]); lcm(v)/gcd(v); \\ Michel Marcus, Aug 31 2020

a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).
a(n) = A336723(n) / A336722(n).
a(p) = p * (p+1) for p = primes.

A334985 a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, 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, 18, 56, 960, 351, 450, 132, 6048, 182, 588, 1800, 158720, 306, 25272, 380, 84000, 14112, 2178, 552, 414720, 11625, 7098, 29160, 32928, 870, 405000, 992, 2064384, 17424, 15606, 58800, 917070336, 1406, 10830, 85176, 11520000, 1722, 3111696

Offset: 1

Jaroslav Krizek, Sep 22 2020

nonn

			a(6) = lcm(6, tau(6), sigma(6), pod(6)) / gcd(6, tau(6), sigma(6), pod(6))  = lcm(6, 4, 12, 36) / gcd(6, 4, 12, 36) = 36 / 2 = 18.

Cf. A336722, A336723, A337323.

Cf. A329929 (lcm(tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n))).

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

a[n_] := LCM @@ {(d = DivisorSigma[0, n]), (s = DivisorSigma[1, n]), n^(d/2)} / GCD @@ {n, d, s}; Array[a, 50] (* Amiram Eldar, Sep 22 2020 *)

a(n) = my(f=factor(n), v=[n, numdiv(f), sigma(f), vecprod(divisors(f))]); lcm(v)/gcd(v); \\ Michel Marcus, Sep 22 2020

a(n) = lcm(tau(n), sigma(n), pod(n)) / gcd(n, tau(n), sigma(n)).

a(n) = A336723(n) / A337323(n).

A337325 a(n) is the smallest number m such that gcd(tau(m), sigma(m), pod(m)) = 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, 10, 18, 6, 5000, 90, 66339, 30, 288, 3240, 10036224, 60, 582160384, 20412, 16200, 168, 49030215219, 612, 4637065216, 1520, 142884, 912384, 98881718827959, 420, 7543125, 479232, 14112, 5824, 26559758051835904, 104400, 25796647321600, 840, 491774976, 1268973568

Offset: 1

Jaroslav Krizek, Aug 23 2020

nonn

p^(q-1) | a(q). If p != q then (p^(q-1) * q) | a(q) for some primes p and q. A similar idea can be used for nonprime q. - David A. Corneth, Aug 25 2020

			For n = 6; a(6) = 90 because 90 is the smallest number with gcd(tau(90), sigma(90), pod(90)) = gcd(12, 234, 531441000000) = 6.

Cf. A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A337324 (least m such that gcd(m,tau(m),sigma(m),pod(m)) = n).
Cf. A000005, A000203, A007955.

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

f(n) = my(d=divisors(factor(n))); gcd([#d, vecsum(d), vecprod(d)]);
a(n) = my(m=1); while (f(m) != n, m++); m; \\ Michel Marcus, Sep 21 2020

a(11) and a(13) from Amiram Eldar, Aug 25 2020

More terms from Jinyuan Wang, Oct 03 2020

A338406 Numbers m such that tau(m), sigma(m) and pod(m) are pairwise relatively prime.

Original entry on oeis.org

1, 4, 16, 25, 64, 81, 100, 121, 256, 289, 484, 529, 729, 841, 1024, 1156, 1296, 1600, 1681, 1936, 2116, 2209, 2401, 2809, 3025, 3364, 3481, 4096, 4624, 5041, 5184, 6400, 6724, 6889, 7225, 7921, 8464, 8836, 10201, 11236, 11449, 11664, 12100, 12769, 13225, 13456

Offset: 1

Jaroslav Krizek, Oct 24 2020

nonn

Numbers m such that A336723(m) = A000005(m) * A000203(m) * A007955(m).
Numbers m such that lcm(m, tau(m), sigma(m), pod(m)) = tau(m) * sigma(m) * pod(m).
Subsequence of numbers m such that A336722(m) = gcd(tau(m), sigma(m), pod(m)) = 1.
From David A. Corneth, Dec 11 2020: (Start)
a(n) is a perfect square. Proof: If a(n) is not a perfect square but is even then both tau(a(n)) and pod(a(n)) are divisible by 2. Contradiction.
If a(n) is not a perfect square and is odd then both tau(a(n)) and sigma(a(n)) are even. Contradiction.
Hence if a(n) is not a perfect square then it can be neither even nor odd. So a(n) is a perfect square. Q.E.D. (End)

			lcm(tau(4), sigma(4), pod(4)) = lcm(3, 7, 8) = tau(4) * sigma(4) * pod(4) = 3 * 7 * 8 = 168.

David A. Corneth, Table of n, a(n) for n = 1..10000

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

Cf. A336722, A336723, A338395.

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

Select[Range[15000], CoprimeQ[(d = DivisorSigma[0, #]), (s = DivisorSigma[1, #])] && CoprimeQ[d, (p = #^(d/2))] && CoprimeQ[s, p] &] (* Amiram Eldar, Oct 25 2020 *)

isok(m) = my(d=divisors(m), t=#d, s=vecsum(d), p=vecprod(d)); t*s*p == lcm([t,s,p]); \\ Michel Marcus, Oct 25 2020

cp's OEIS Frontend

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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A337323 a(n) = gcd(n, 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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A329929 a(n) = lcm(tau(n), sigma(n), pod(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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A334985 a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, 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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A337325 a(n) is the smallest number m such that gcd(tau(m), sigma(m), pod(m)) = 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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A338406 Numbers m such that tau(m), sigma(m) and pod(m) are pairwise relatively prime.

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