cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Jason Earls, May 19 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

Extensions

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

A163080 Primes p such that p$ - 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

3, 5, 7, 13, 41, 47, 83, 137, 151, 229, 317, 389, 1063, 2371, 6101, 7873, 13007, 19603
Offset: 1

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Comments

a(n) are the primes in A163078.

Examples

			3 is prime and 3$ - 1 = 5 is prime, so 3 is in the sequence.

Links

Crossrefs

Cf. A056040, A103317, A163079, A163078.

Programs

  • Maple
    a := proc(n) select(isprime,select(k -> isprime(A056040(k)-1),[$0..n])) end:
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2; Select[Prime /@ Range[200], PrimeQ[sf[#] - 1] &] (* Jean-François Alcover, Jun 28 2013 *)
  • PARI
    is(k) = isprime(k) && ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020

Extensions

a(14)-a(18) from Jinyuan Wang, Mar 22 2020

A331547 Numbers k such that k and k! - 1 have the same number of divisors.

Original entry on oeis.org

3, 7, 8, 10, 26, 27, 34, 85, 93, 104, 143, 152
Offset: 1

Views

Author

Matthew Niemiro, Jan 20 2020

Keywords

Comments

The sequence also includes: 143, 152, 186, 230, 379, 381, 543, 573, 602. - Daniel Suteu, Jan 21 2020
The sequence also includes 2881. Even though the complete factorization of 136!-1 is not known, 136 is not a term, since 136!-1 is known to be the product of 2 distinct primes and a composite number, so it has at least 4 prime factors and 3 distinct prime factors, thus the number of divisors >= 12, whereas 136 has 8 divisors. - Chai Wah Wu, Feb 26 2020
Similar reasoning (considering only small prime factors of k! - 1) shows that the next terms (> a(12) = 152) can only be within the set {154, 160, 162, 164, 176, 180, 182, 186, 187, 188, 192, 195, 196, 198, 204, ...}. - M. F. Hasler, Feb 26 2020

Links

Crossrefs

Supersequence of A103317.
Cf. A000005, A002982, A064145.

Programs

  • Mathematica
    Select[Range[50], DivisorSigma[0, #] - DivisorSigma[0, Factorial[#] - 1] == 0 &]
  • PARI
    isok(k) = k>1 && numdiv(k)==numdiv(k!-1); \\ Jinyuan Wang, Jan 20 2020
  • PARI
    {is(n)=my(f); n>2&& numdiv(n)>=numdiv(f=factor(n!-1,0))&& if( ispseudoprime(vecmax(f[,1])), numdiv(n)==numdiv(f), numdiv(n)<2*numdiv(f), 0, numdiv(n)==numdiv(n!-1))} \\ Avoids complete factorization if possible. - M. F. Hasler, Feb 26 2020

Formula

A331547 = { n > 1 | A000005(n) = A064145(n) }. - M. F. Hasler, Feb 26 2020

Extensions

a(8)-a(9) from Jinyuan Wang, Jan 20 2020
a(10) from Amiram Eldar, Jan 20 2020
a(11)-a(12) from Chai Wah Wu, Feb 26 2020

A110094 Startorial primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 1451521, 2903041, 5806081, 46448639, 92897281, 371589121, 10032906239, 30098718719, 270888468479, 812665405439, 7313988648961, 21941965946881, 89874292518420479
Offset: 1

Views

Author

Jonathan Vos Post, Sep 04 2005

Keywords

Comments

These are primes of the form A109834 startorials (base 10) +1 or -1. This is by analogy to factorial primes (A002981), superfactorial primes (A073828), hyperfactorial primes, ultrafactorial primes (comment in A046882), subfactorial primes (A100015), double factorial primes (A080778), multifactorial primes (A037083).

Crossrefs

Cf. A000040, A002981, A073828, A080778, A037083, A100015, A103317, A046882, A109834.

Formula

{a(n)} = {A109834(k)+1 an element of A000040, or A109834(k)-1 an element of A000040, for some k}.

A122724 Primes p such that (2p)! - 1 is also prime.

Original entry on oeis.org

2, 3, 7, 19, 47, 83, 487
Offset: 1

Views

Author

Alexander Adamchuk, Sep 23 2006

Keywords

Comments

Corresponding primes of the form (2p)! - 1 are {23,719,87178291199,523022617466601111760007224100074291199999999,...}.
No other terms < 20663. - Robert Price, Mar 02 2012
No other terms < 104002 using A002982. - Michael S. Branicky, May 14 2025

Crossrefs

Cf. A002982, A103317.

Programs

  • Mathematica
    Select[Prime[Range[100]],PrimeQ[(2#)!-1]&] (* James C. McMahon, Nov 09 2024 *)
Showing 1-5 of 5 results.

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