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 1-7 of 7 results.

A063499 Primes of the form prime(n) + n!.

Original entry on oeis.org

3, 5, 11, 31, 131, 733, 362903, 39916831, 355687428096059, 6402373705728061, 15511210043330985984000097, 8222838654177922817725562880000127, 815915283247897734345611269596115894272000000173
Offset: 1

Views

Author

Jason Earls, Jul 30 2001

Keywords

Comments

Subsequence of A121926. - Michel Marcus, Apr 05 2015

Links

Crossrefs

Cf. A064278 (Numbers n such that n! + prime(n) is prime). [From Alexander R. Povolotsky, Aug 13 2008]

Programs

  • Magma
    [a: n in [1..50] | IsPrime(a) where a is NthPrime(n) + Factorial(n) ]; // Vincenzo Librandi, Apr 05 2015
  • Mathematica
    Select[Table[Prime[n] + n!, {n, 1, 60}], PrimeQ] (* Vincenzo Librandi, Apr 05 2015 *)
  • PARI
    for(n=1,70,x=prime(n)+n!; if(isprime(x),print(x)))
  • PARI
    { n=0; f=1; for (m=1, 10^9, f*=m; if (isprime(a=prime(m) + f), write("b063499.txt", n++, " ", a); if (n==18, break)) ) } \\ Harry J. Smith, Aug 24 2009

A064401 Numbers k such that k! - prime(k) is prime.

Original entry on oeis.org

4, 5, 7, 12, 14, 15, 16, 17, 18, 20, 30, 37, 39, 49, 52, 66, 86, 162, 165, 202, 235, 250, 366, 419, 1169, 1311, 1916, 3032, 3211, 3335, 4650, 6199, 7762, 12776
Offset: 1

Views

Author

Robert G. Wilson v, Sep 28 2001

Keywords

Comments

a(34) > 12500. - Giovanni Resta, Mar 16 2014

Crossrefs

Cf. A064278.

Programs

  • Magma
    [n: n in [1..200] | IsPrime(Factorial(n)- NthPrime(n))]; // Vincenzo Librandi, Mar 05 2015
  • Mathematica
    Do[ If[ PrimeQ[ n! - Prime[ n ] ], Print[ n ] ], {n, 1, 600} ]
Select[Range[2000], PrimeQ[#! - Prime[#]] &] (* Vincenzo Librandi, Mar 05 2015 *)

Extensions

a(25)-a(27) from Farideh Firoozbakht, Feb 28 2004
a(28)-a(33) from Giovanni Resta, Mar 16 2014
a(34) from Michael S. Branicky, Apr 20 2025

A084749 Numbers m such that m! + p is a prime, where p is the smallest prime > m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 33, 44, 48, 52, 64, 73, 92, 119, 182, 487, 603, 987, 4884, 6822, 8070, 11079, 13659, 17659
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 16 2003

Keywords

Comments

Next term, if it exists, is >4800. - Ryan Propper, Jan 02 2007
From Farideh Firoozbakht, Oct 21 2009: (Start)
Numbers corresponding to a(19)-a(24) are probable primes.
There is no further term up to 8300. (End)

Examples

			727 = 6! + 7 is a prime but 8! + 11 is composite hence 6 is a member but 8 is not.
7 is in the sequence because 7!=5040, nextprime(7)=11 and 5040+11 is prime.

Crossrefs

Cf. A084748, A084750, A092028, A064278, A002981, A151892.

Programs

  • Mathematica
    Do[If[PrimeQ[k!+NextPrime[k]], Print[k]], {k, 0, 1525}] (* Farideh Firoozbakht, Feb 26 2004 *)
Select[Range[0,500],PrimeQ[#!+NextPrime[#]]&] (* The program generates the first 19 terms of the sequence. *) (* Harvey P. Dale, Jul 16 2025 *)

Extensions

More terms from Farideh Firoozbakht, Feb 26 2004
Edited by N. J. A. Sloane at the suggestion of Artur Jasinski, Apr 14 2008
a(22)-a(24) from Farideh Firoozbakht, Oct 21 2009
a(25) from Michael S. Branicky, Aug 05 2024
a(26)-a(27) from Michael S. Branicky, May 25 2025

A236263 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 3, 3, 4, 5, 4, 4, 5, 7, 4, 5, 6, 6, 5, 5, 5, 7, 6, 7, 9, 7, 8, 7, 7, 5, 11, 8, 8, 8, 11, 8, 7, 5, 10, 6, 9, 8, 10, 7, 8, 10, 9, 7, 8, 9, 13, 8, 8, 9, 10, 6, 11, 10, 7, 7, 9, 11, 13, 8, 11, 13, 11, 14, 6
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 21 2014

Keywords

Comments

It seems that a(n) > 0 for all n > 17. (We have verified this for n up to 13000.) If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.
See also A236265 for a similar sequence.

Examples

			a(18) = 1 since phi(3)/2 + phi(15)/8 = 1 + 1 = 2 with 2! + prime(2) = 2 + 3 = 5 prime.
a(356) = 1 since phi(203)/2 + phi(153)/8 = 84 + 12 = 96 with 96! + prime(96) = 96! + 503 prime.
a(457) = 1 since phi(7)/2 + phi(450)/8 = 3 + 15 = 18 with 18! + prime(18) = 18! + 61 = 6402373705728061 prime.

Links

Crossrefs

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236265.

Programs

  • Mathematica
    q[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236265 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! - prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 2, 2, 4, 3, 5, 1, 3, 2, 3, 3, 4, 5, 9, 5, 5, 6, 7, 8, 8, 8, 5, 7, 5, 8, 8, 5, 5, 9, 8, 6, 6, 9, 8, 10, 6, 9, 4, 6, 9, 9, 8, 10, 9, 6, 10, 7, 8, 12, 11, 10, 8, 11, 9, 12, 7, 13, 12, 13
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 21 2014

Keywords

Comments

It seems that a(n) > 0 for all n > 21. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! - prime(m) prime.
See also A236263 for a similar sequence.

Examples

			a(23) = 1 since phi(7)/2 + phi(16)/8 = 3 + 1 = 4 with 4! - prime(4) = 24 - 7 = 17 prime.
a(26) = 1 since phi(9)/2 + phi(17)/8 = 3 + 2 = 5 with 5! - prime(5) = 120 - 11 = 109 prime.

Links

Crossrefs

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263.

Programs

  • Mathematica
    q[n_]:=IntegerQ[n]&&PrimeQ[n!-Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236325 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 2, 3, 4, 3, 4, 4, 5, 2, 4, 3, 4, 5, 5, 6, 5, 6, 8, 7, 9, 8, 6, 6, 5, 8, 9, 4, 8, 7, 7, 5, 5, 7, 7, 8, 8, 6, 7, 8, 7, 10, 5, 8, 9, 8, 7, 7, 6, 7, 8, 12, 10, 6, 8, 9, 9, 12, 9, 8, 7, 13
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 22 2014

Keywords

Comments

It might seem that a(n) > 0 for all n > 14, but a(7365) = 0. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.

Examples

			a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 1! + prime(1) = 1 + 2 = 3 prime.
a(23) = 1 since phi(10)/2 + phi(13)/12 = 2 + 1 = 3 with 3! + prime(3) = 6 + 5 = 11 prime.

Links

Crossrefs

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263, A236265.

Programs

  • Mathematica
    p[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A264723 Primes of the form n! - prime(n).

Original entry on oeis.org

17, 109, 5023, 479001563, 87178291157, 1307674367953, 20922789887947, 355687428095941, 6402373705727939, 2432902008176639929, 265252859812191058636308479999887, 13763753091226345046315979581580902399999843, 20397882081197443358640281739902897356799999833
Offset: 1

Views

Author

Vincenzo Librandi, Nov 22 2015

Keywords

Comments

Subsequence of A261809. - Altug Alkan, Nov 22 2015

Crossrefs

Cf. A063499, A064278, A064401, A261809.

Programs

  • Magma
    [a: n in [1..40] | IsPrime(a) where a is Factorial(n)-NthPrime(n)];
  • Mathematica
    Select[Table[n! - Prime[n], {n, 50}], PrimeQ]
  • PARI
    for(n=1, 1e2, if(isprime(k=(n!-prime(n))), print1(k, ", "))) \\ Altug Alkan, Nov 22 2015
Showing 1-7 of 7 results.

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