A062569 -id:A062569 - cp's OEIS frontend

A275369 Least k such that n! divides sigma(k!) (k > 0).

1, 3, 3, 5, 5, 8, 14, 19, 19, 23, 23, 30, 30, 31, 50, 50, 50, 50, 50, 51, 51, 64, 90, 90, 91, 91, 91, 91, 126, 130, 130, 130, 130, 130, 130, 130, 130, 130, 131, 132, 132, 132, 132, 132, 134, 134, 234, 234, 234, 234, 236, 236, 236, 236, 288, 288, 288, 288

Offset: 1

Altug Alkan, Jul 29 2016

nonn

			a(3) = 3 because 3! divides sigma(3!) = 12.

Robert Israel, Table of n, a(n) for n = 1..1100

Cf. A000142, A062569.

N:= 500: # to get a(1) .. a(N)
k:= 1; skf:= 1;
for n from 1 to N do
  nf:= n!;
  while skf mod nf <> 0 do
    k:= k+1;
    skf:= numtheory:-sigma(k!);
  od:
  A[n]:= k;
od:
seq(A[i],i=1..N); # Robert Israel, Aug 09 2016

Table[k = 1; While[! Divisible[DivisorSigma[1, k!], n!], k++]; k, {n, 58}] (* Michael De Vlieger, Aug 08 2016 *)

a(n) = {my(k = 1); while(sigma(k!) % n! != 0, k++); k; }

A275769 Least k such that n divides sigma(k!) (k > 0).

Original entry on oeis.org

1, 3, 2, 3, 4, 3, 10, 5, 5, 4, 9, 3, 6, 10, 4, 7, 8, 5, 14, 4, 11, 9, 12, 5, 21, 6, 14, 10, 42, 4, 6, 11, 9, 8, 14, 5, 36, 14, 6, 5, 22, 11, 35, 9, 5, 12, 24, 7, 13, 21, 8, 7, 69, 14, 9, 10, 14, 42, 60, 4, 21, 6, 14, 12, 8, 9, 45, 8, 12, 14, 20, 5, 10, 36, 21, 14, 10, 6, 40, 8, 14, 22, 81, 11

Offset: 1

Altug Alkan, Aug 08 2016

nonn

			a(5) = 4 because sigma(4!) = 60 is divisible by 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000203, A062569.

Table[k = 1; While[! Divisible[DivisorSigma[1, k!], n], k++]; k, {n, 84}] (* Michael De Vlieger, Aug 08 2016 *)
Module[{nn=100,sk},sk=Table[{k,DivisorSigma[1,k!]},{k,nn}];Table[SelectFirst[sk,Mod[#[[2]],n]==0&],{n,nn}]][[;;,1]] (* Harvey P. Dale, Nov 28 2024 *)

a(n) = {my(k=1); while(sigma(k!) % n != 0, k++); k; }

A294346 E.g.f.: exp( Sum_{n>=1} sigma(n!) * x^n/n! ).

Original entry on oeis.org

1, 1, 4, 22, 154, 1306, 12874, 145954, 1843660, 25840684, 397040064, 6637371896, 119517187984, 2310108276048, 47619441310520, 1042743337601320, 24164137431011184, 590726322945970352, 15184954152657360064, 409428979786326488096, 11550423660014156192096, 340219279585618435264480, 10442779307230643663779424, 333425628200639984852617568, 11055838405832227887079632832

Offset: 0

Paul D. Hanna, Oct 29 2017

nonn

Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 154*x^4/4! + 1306*x^5/5! + 12874*x^6/6! + 145954*x^7/7! + 1843660*x^8/8! + 25840684*x^9/9! + 397040064*x^10/10! + 6637371896*x^11/11! + 119517187984*x^12/12! +...
such that
log(A(x)) = x + sigma(2!)*x^2/2! + sigma(3!)*x^3/3! + sigma(4!)*x^4/4! + sigma(5!)*x^5/5! + sigma(6!)*x^6/6! +...+ sigma(n!)*x^n/n! +...
Explicitly,
log(A(x)) = x + 3*x^2/2! + 12*x^3/3! + 60*x^4/4! + 360*x^5/5! + 2418*x^6/6! + 19344*x^7/7! + 159120*x^8/8! + 1481040*x^9/9! + 15334088*x^10/10! + 184009056*x^11/11! + 2217441408*x^12/12! +...+ A062569(n)*x^n/n! +...
PRODUCT.
A(x) = 1 / ((1-x) * (1-x^2)^(2/2!) * (1-x^3)^(10/3!) * (1-x^4)^(42/4!) * (1-x^5)^(336/5!) * (1-x^6)^(1458/6!) * (1-x^7)^(18624/7!) * (1-x^8)^(108720/8!) * (1-x^9)^(1239120/9!) * (1-x^10)^(9165128/10!) * (1-x^11)^(180380256/11!) * (1-x^12)^(1133700288/12!) *...).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A062569.

{a(n) = n!*polcoeff( exp( sum(m=1, n+1, sigma(m!) * x^m/m!) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

A167369 sigma(n!,n!).

Original entry on oeis.org

1, 5, 47450, 1333735856351858432985890996140258

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 01 2009

nonn

Cf. A062569, A167367, A167368

Mathematica
```
Array[DivisorSigma[ #!,#! ]&,5,1]
```

A306507 a(n) = gcd(n!^2+1, sigma(n!)), where sigma() denotes the sum of the divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 13, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 1, 1, 193, 1, 1, 1, 757, 61, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 109, 1, 1, 1, 181, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 113

Offset: 1

Daoudi Rédoane, Feb 20 2019

nonn

A sequence that produces primes.
A counterexample is found at n=7880, here the gcd is 380927609 = 15761*24169.
Interesting properties may be found in this sequence, for example many primes are 2n+1.

Cf. A000203, A020549, A062569.

Table[GCD[(n!)^2+1,DivisorSigma[1,n!]],{n,90}] (* Harvey P. Dale, Jun 03 2021 *)

a(n) = gcd(n!^2+1, sigma(n!)); \\ Michel Marcus, Feb 20 2019

a(n) = gcd(A020549(n), A062569(n)).

More terms from Michel Marcus, Feb 20 2019

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A275369 Least k such that n! divides sigma(k!) (k > 0).

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A275769 Least k such that n divides sigma(k!) (k > 0).

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A294346 E.g.f.: exp( Sum_{n>=1} sigma(n!) * x^n/n! ).

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A167369 sigma(n!,n!).

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A306507 a(n) = gcd(n!^2+1, sigma(n!)), where sigma() denotes the sum of the divisors.

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