A062363 -id:A062363 - cp's OEIS frontend

A062795 Duplicate of A062363.

0, 1, 3, 7, 27, 121, 729, 5041, 40347, 362887, 3628923, 39916801, 479002353

Offset: 0

dead

A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).

1, 1, 2, 4, 11, 37, 167, 925, 6164, 47630, 418227, 4105887, 44529413, 528398441, 6807143686, 94588353184, 1409913624333, 22437692156739, 379673925360239, 6806484898946045, 128862141334488784, 2569079946351669286, 53797816061915662161, 1180533553597621952193

Offset: 0

Paul D. Hanna, Jan 08 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 167*x^6 +...
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)^6*(1-x^5)^24*(1-x^6)^120*...).
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 121*x^5/5 + 729*x^6/6 + 5041*x^7/7 + 40347*x^8/8 +...+ A062363(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A062363, A107895, A256126, A261047.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial((i-1)!+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 10 2021

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k-1)!),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d!)*x^m/m)+x*O(x^n)),n)}

Euler transform of (n-1)!.
G.f.: A(x) = exp( Sum_{n>=1} A062363(n)*x^n/n ) where A062363(n) = Sum_{d|n} d!.
a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1860/n^6 + 14793/n^7 + 138166/n^8 + 1494034/n^9 + 18422609/n^10), for coefficients see A256126. - Vaclav Kotesovec, Mar 14 2015

A093804 Primes p such that p! + 1 is also prime.

Original entry on oeis.org

2, 3, 11, 37, 41, 73, 26951, 110059, 150209

Offset: 1

Jason Earls, May 19 2004

Or, numbers n such that Sum_{d|n} d! is prime.
The prime 26951 from A002981 (n!+1 is prime) is a member since Sum_{d|n} d! = n!+1 if n is prime. - Jonathan Sondow, Jan 30 2005
a(n) are the primes in A002981[n] = {0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ...} Numbers n such that n! + 1 is prime. Corresponding primes of the form p! + 1 are listed in A103319[n] = {3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, ...}. - Alexander Adamchuk, Sep 23 2006

			Sum_{d|3} d! = 1! + 3! = 7 is prime, so 3 is a member.

Chris K. Caldwell, The List of Largest Known Primes, 110059! + 1
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
Apoloniusz Tyszka, A common approach to the problem of the infinitude of twin primes, primes of the form n! + 1, and primes of the form n! - 1, 2018.
Apoloniusz Tyszka, A new approach to solving number theoretic problems, 2018.

Cf. A062363, A002981, A038507, A088332, A103317, A103319.

seq(`if`(isprime(ithprime(n)!+1), ithprime(n), NULL),n=1..25); # Nathaniel Johnston, Jun 28 2011

Select[Prime[Range[5! ]],PrimeQ[ #!+1]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

isok(n) = ispseudoprime(n) && ispseudoprime(n!+1); \\ Jinyuan Wang, Jan 20 2020

One more term from Alexander Adamchuk, Sep 23 2006
a(8)=110059 (found on Jun 11 2011, by PrimeGrid), added by Arkadiusz Wesolowski, Jun 28 2011
a(9)=150209 (found on Jun 09 2012, by Rene Dohmen), added by Jinyuan Wang, Jan 20 2020

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 5, 31, 67, 139, 631, 9743, 16253, 17977, 27901, 37589

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163077.

			5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A062363, A093804, A163077.

Cf. A056040. - Robert G. Wilson v, Aug 09 2010

a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* Robert G. Wilson v, Aug 08 2010 *)

is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(8)-a(12) from Robert G. Wilson v, Aug 08 2010

A217576 a(n) = Sum_{d divides n} (d!)^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 765, 5041, 40913, 363097, 3643233, 39916801, 479535185, 6227020801, 87203692929, 1307676103777, 20924415922433, 355687428096001, 6402505760917569, 121645100408832001, 2432915176581403649, 51090942299733783937, 1124002321128529922049

Offset: 1

Joerg Arndt, Oct 07 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A062363 ( Sum_{d divides n} d! ).

Cf. A062796 ( Sum_{d divides n} d^d ), A066108 ( Sum_{d divides n} n^d ).

f[n_]=With[{d=Divisors[n]},Total[(d!)^(n/d)]]; Array[f,25] (* Harvey P. Dale, Dec 20 2023 *)

PARI
```
a(n)=sumdiv(n,d, (d!)^(n/d) );
```

G.f.: Sum_{n>=1} n!*x^n / (1 - n!*x^n). - Paul D. Hanna, Jan 17 2013

A348146 a(n) = Sum_{d|n} (d!)^(n-d).

Original entry on oeis.org

1, 2, 2, 6, 2, 234, 2, 331842, 46658, 24883200258, 2, 139314179589392898, 2, 82606411253903523840004098, 619173642242176782338, 6984964247141514123665660725036072962, 2, 109110688415571335888754861121236891599318185050114, 2

Offset: 1

Wesley Ivan Hurt, Oct 02 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..61

Cf. A062363, A342628, A345465, A346196.

Table[Sum[(i!)^(n - i) (1 - Ceiling[n/i] + Floor[n/i]), {i, n}], {n, 20}]

a(n) = sumdiv(n, d, d!^(n-d)); \\ Seiichi Manyama, Oct 03 2021

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k!*x)^k))) \\ Seiichi Manyama, Oct 03 2021

a(p) = 2 for primes p.

G.f.: Sum_{k>=1} x^k/(1 - (k! * x)^k). - Seiichi Manyama, Oct 03 2021

A351708 a(n) = Sum_{p|n, p prime} (n/p)!.

Original entry on oeis.org

0, 1, 1, 2, 1, 8, 1, 24, 6, 122, 1, 744, 1, 5042, 126, 40320, 1, 363600, 1, 3628824, 5046, 39916802, 1, 479041920, 120, 6227020802, 362880, 87178291224, 1, 1307677997520, 1, 20922789888000, 39916806, 355687428096002, 5160, 6402374184729600, 1

Offset: 1

Wesley Ivan Hurt, Feb 16 2022

			a(6) = 8; a(6) = Sum_{p|6} (6/p)! = (6/2)! + (6/3)! = 3*2 + 2*1 = 8.

Robert Israel, Table of n, a(n) for n = 1..897

Cf. A000040, A000142, A062363, A351709.

f:= proc(n) local p;
  add((n/p)!, p = numtheory:-factorset(n))
end proc:
map(f, [$1..40]); # Robert Israel, Dec 05 2022

Table[DivisorSum[n, Factorial[n/#] &, PrimeQ], {n, 36}] (* Michael De Vlieger, Dec 06 2022 *)

a(A000040(n)) = 1.

A321521 a(n) = Sum_{d|n} (-1)^(n/d+1)*d!.

Original entry on oeis.org

1, 1, 7, 21, 121, 715, 5041, 40293, 362887, 3628681, 39916801, 479000895, 6227020801, 87178286161, 1307674368127, 20922789847653, 355687428096001, 6402373705365835, 121645100408832001, 2432902008173011101, 51090942171709445047, 1124000727777567763201

Offset: 1

Ilya Gutkovskiy, Nov 12 2018

nonn

Cf. A000142, A062363, A321522.

Table[Sum[(-1)^(n/d + 1) d!, {d, Divisors[n]}], {n, 22}]
nmax = 22; Rest[CoefficientList[Series[Sum[k! x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d!); \\ Michel Marcus, Nov 12 2018

G.f.: Sum_{k>=1} k!*x^k/(1 + x^k).

a(n) ~ n!. - Vaclav Kotesovec, Nov 13 2018

A321875 a(n) = Sum_{d|n} d*d!.

Original entry on oeis.org

1, 5, 19, 101, 601, 4343, 35281, 322661, 3265939, 36288605, 439084801, 5748023639, 80951270401, 1220496112085, 19615115520619, 334764638530661, 6046686277632001, 115242726706374263, 2311256907767808001, 48658040163569088701, 1072909785605898275299

Offset: 1

Ilya Gutkovskiy, Nov 20 2018

nonn

Inverse Möbius transform of A001563.

Seiichi Manyama, Table of n, a(n) for n = 1..448
N. J. A. Sloane, Transforms

Cf. A000142, A001563, A062363, A107895.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k*Factorial(k)*x^k/(1 - x^k): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 20 2018

Table[Sum[d d!, {d, Divisors[n]}], {n, 21}]
nmax = 21; Rest[CoefficientList[Series[Sum[k k! x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 21; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, d*d!); \\ Michel Marcus, Nov 20 2018

s = sum(k*factorial(k)*x^k/(1-x^k) for k in (1..24));
(s/x).series(x, 21).coefficients(x, sparse=false) # Peter Luschny, Nov 21 2018

G.f.: Sum_{k>=1} k*k!*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k!)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} A001563(d).

A351709 a(n) = Sum_{p|n, p prime} p!.

Original entry on oeis.org

0, 2, 6, 2, 120, 8, 5040, 2, 6, 122, 39916800, 8, 6227020800, 5042, 126, 2, 355687428096000, 8, 121645100408832000, 122, 5046, 39916802, 25852016738884976640000, 8, 120, 6227020802, 6, 5042, 8841761993739701954543616000000, 128, 8222838654177922817725562880000000

Offset: 1

Wesley Ivan Hurt, Feb 16 2022

nonn

Inverse Möbius transform of n! * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

			a(6) = 8; a(6) = Sum_{p|6} p! = 2! + 3! = 2*1 + 3*2*1 = 8.

Cf. A000040, A000142, A010051, A062363, A351708.

a(n) = Sum_{d|n} d! * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

cp's OEIS Frontend

A062795 Duplicate of A062363.

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A179327 G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).

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A093804 Primes p such that p! + 1 is also prime.

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A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

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A217576 a(n) = Sum_{d divides n} (d!)^(n/d).

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A348146 a(n) = Sum_{d|n} (d!)^(n-d).

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A351708 a(n) = Sum_{p|n, p prime} (n/p)!.

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A321521 a(n) = Sum_{d|n} (-1)^(n/d+1)*d!.

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