A009278 -id:A009278 - cp's OEIS frontend

A009205 a(n) = gcd(d(n), sigma(n)).

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

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A009191, A009194, A009278, A064840, A087801, A286360.

Table[GCD[DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Dec 05 2017 *)

A009205(n) = gcd(numdiv(n),sigma(n)); \\ Antti Karttunen, May 22 2017

from math import prod, gcd
from sympy import factorint
def A009205(n):
    f = factorint(n).items()
    return gcd(prod(e+1 for p, e in f),prod((p**(e+1)-1)//(p-1) for p,e in f)) # Chai Wah Wu, Jul 27 2023

a(n) = A064840(n)/A009278(n). - Amiram Eldar, Jan 31 2025

Data section extended to 120 terms by Antti Karttunen, May 22 2017

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).

A334784 a(n) = Sum_{d|n} lcm(tau(d), sigma(d)).

Original entry on oeis.org

1, 7, 5, 28, 7, 23, 9, 88, 44, 49, 13, 128, 15, 39, 35, 243, 19, 140, 21, 112, 45, 55, 25, 308, 100, 105, 84, 228, 31, 161, 33, 369, 65, 133, 63, 1064, 39, 87, 75, 532, 43, 183, 45, 160, 152, 103, 49, 1083, 66, 328, 95, 420, 55, 300, 91, 408, 105, 217, 61, 476

Offset: 1

Jaroslav Krizek, May 10 2020

nonn

			a(6) = lcm(tau(1), sigma(1)) + lcm(tau(2), sigma(2)) + lcm(tau(3), sigma(3)) + lcm(tau(6), sigma(6)) = lcm(1, 1) + lcm(2, 3) + lcm(2, 4) + lcm(4, 12) = 1 + 6 + 4 + 12 = 23.

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334783 (Sum_{d|n} lcm(d, sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009278 (lcm(tau(n), sigma(n))).

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

a[n_] := DivisorSum[n, LCM[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)

a(n) = sumdiv(n, d, lcm(numdiv(d), sigma(d))); \\ Michel Marcus, May 10 2020

a(p) = p + 2 for p = odd primes (A065091).

A307740 Numbers k such that k divides lcm(tau(k), sigma(k)).

Original entry on oeis.org

1, 2, 6, 12, 24, 28, 40, 84, 120, 252, 360, 496, 672, 2480, 3276, 4680, 7440, 8128, 30240, 32760, 56896, 293760, 435708, 523776, 997920, 2178540, 2618880, 8910720, 23569920, 33550336, 45532800, 64995840, 102136320, 142990848, 275890944, 436154368, 459818240

Offset: 1

Jaroslav Krizek, Apr 26 2019

nonn

Numbers k such that k divides A009278(k).
Conjecture: multiply-perfect numbers (A007691) are terms.
Corresponding values of lcm(tau(k), sigma(k)) / k for numbers k from this sequence: 1, 3, 2, 7, 5, 6, 9, 8, 6, 26, 13, 10, 3, 12, 28, 14, 16, 14, 4, ...
Sequence of the smallest numbers k such that lcm(tau(k), sigma(k)) / k = n for n >= 1: 1, 6, 2, 30240, 24, 28, 12, 84, 40, 496, ...

			6 is in the sequence because lcm(tau(6), sigma(6)) / 6 = lcm(4, 12) / 6 = 12 / 6 = 2.

Cf. A000005, A000203, A007691, A009278.

[n: n in [1..1000000] | LCM(SumOfDivisors(n), NumberOfDivisors(n)) mod n eq 0]

aQ[n_]:=Divisible[LCM@@(DivisorSigma[#,n]&/@{0,1}),n]; Select[Range[10000], aQ] (* Amiram Eldar, May 07 2019 *)

isok(n) = !(lcm(numdiv(n), sigma(n)) % n); \\ Michel Marcus, Apr 26 2019

More terms from Amiram Eldar, May 07 2019

A334806 a(n) = Product_{d|n} lcm(tau(d), sigma(d)) where tau(k) is the number of divisors of k (A000005) and sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 4, 126, 6, 288, 8, 7560, 156, 1296, 12, 508032, 14, 1152, 576, 1171800, 18, 876096, 20, 1143072, 1024, 2592, 24, 3657830400, 558, 7056, 6240, 4064256, 30, 107495424, 32, 147646800, 2304, 11664, 2304, 1265709908736, 38, 7200, 3136, 24690355200, 42

Offset: 1

Jaroslav Krizek, Jun 26 2020

nonn

			a(6) = lcm(tau(1), sigma(1)) * lcm(tau(2), sigma(2)) * lcm(tau(3), sigma(3)) * lcm(tau(6), sigma(6)) = lcm(1, 1) * lcm(2, 3) * lcm(2, 4) * lcm(4, 12) = 1 * 6 * 4 * 12 = 288.

Cf. A334784 (Sum_{d|n} lcm(tau(d), sigma(d))), A334729 (Product_{d|n} gcd(tau(d), sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009278 (lcm(tau(n), sigma(n))).

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

a[n_] := Product[LCM[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 41] (* Amiram Eldar, Jun 27 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(numdiv(d[k]), sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

a(p) = p + 1 for p = odd primes (A065091).

cp's OEIS Frontend

A009205 a(n) = gcd(d(n), sigma(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A334784 a(n) = Sum_{d|n} lcm(tau(d), sigma(d)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A307740 Numbers k such that k divides lcm(tau(k), sigma(k)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A334806 a(n) = Product_{d|n} lcm(tau(d), sigma(d)) where tau(k) is the number of divisors of k (A000005) and sigma(k) is the sum of divisors of k (A000203).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula