A064401 -id:A064401 - cp's OEIS frontend

A084750 Numbers k such that k! - p is a prime, where p is the smallest prime > k.

4, 5, 10, 11, 12, 14, 29, 53, 81, 90, 116, 236, 323, 346, 1172, 2957, 8400, 14906

Offset: 1

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

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

If k != 3, there does not exist a prime p and a number k such that k! - NextPrime(k) < p < k! - 1. - Farideh Firoozbakht, Feb 26 2004

			10 is in the sequence because 10! = 3628800, NextPrime(10) = 11 and 3628800 - 11 = 3628789 is prime.

Cf. A084749, A050299, A002982, A064401, A084751.

Do[If[PrimeQ[k!-NextPrime[k]], Print[k]], {k, 0, 1425}] (* Farideh Firoozbakht, Feb 26 2004 *)

More terms from Farideh Firoozbakht, Feb 26 2004
a(16) from Ryan Propper, Jul 09 2005
Edited by N. J. A. Sloane at the suggestion of Ryan Propper, Jan 26 2008
a(17) from Michael S. Branicky, Jun 21 2023
a(18) from Michael S. Branicky, Apr 28 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

Zhi-Wei Sun, Jan 21 2014

nonn

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.

			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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8100

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

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

Zhi-Wei Sun, Jan 21 2014

nonn

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.

			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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

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

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

Zhi-Wei Sun, Jan 22 2014

nonn

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.

			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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

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

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}]

A108420 Numbers k such that k!! - prime(k) is prime.

Original entry on oeis.org

14, 20, 54, 56, 144, 206, 212, 436, 1610, 4450, 4512, 5202, 6684, 14318

Offset: 1

Amineh Farzannia (afarzannia(AT)yahoo.com), Jul 06 2005

There is no further term up to 8800. - Farideh Firoozbakht, Aug 19 2005

			14 is a term since 14!! - prime(14) = 645120 - 43 = 645077 is prime.

Cf. A006882, A007749, A064401.

Select[Range[1610], PrimeQ[ #!! - Prime[ # ]] &]

More terms from Farideh Firoozbakht, Aug 19 2005

Name corrected, term 3 removed, and a(14) from Michael S. Branicky, Jan 03 2025

A143713 Numbers k such that k! - prime(k-1) is prime.

Original entry on oeis.org

3, 4, 5, 6, 10, 22, 31, 92, 174, 237, 886, 1075, 1357, 2428, 3700, 6319, 9116

Offset: 1

Alexander R. Povolotsky, Aug 29 2008

Cf. A064401.

Select[Range[2,1000],PrimeQ[#!-Prime[#-1]]&] (* Harvey P. Dale, Oct 31 2013 *)

for(n=2,1000, if(isprime(n! - prime(n-1)), print1(n, ", ")))

a(11) from Harvey P. Dale, Oct 31 2013
a(12)-a(15) from Metin Sariyar, Sep 27 2019
a(16) from Michael S. Branicky, Jun 08 2023
a(17) from Michael S. Branicky, Apr 23 2025

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

Vincenzo Librandi, Nov 22 2015

nonn

Subsequence of A261809. - Altug Alkan, Nov 22 2015

Cf. A063499, A064278, A064401, A261809.

[a: n in [1..40] | IsPrime(a) where a is Factorial(n)-NthPrime(n)];

Select[Table[n! - Prime[n], {n, 50}], PrimeQ]

for(n=1, 1e2, if(isprime(k=(n!-prime(n))), print1(k, ", "))) \\ Altug Alkan, Nov 22 2015

cp's OEIS Frontend

A084750 Numbers k such that k! - p is a prime, where p is the smallest prime > k.

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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.

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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.

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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.

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A108420 Numbers k such that k!! - prime(k) is prime.

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A143713 Numbers k such that k! - prime(k-1) is prime.

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PARI

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

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PARI