A063499 -id:A063499 - cp's OEIS frontend

A064278 Numbers k such that k! + prime(k) is prime.

1, 2, 3, 4, 5, 6, 9, 11, 17, 18, 25, 31, 40, 96, 174, 193, 204, 269, 523, 650, 659, 797, 1437, 1862, 2515, 4983, 5557, 11429

Offset: 1

Jason Earls, Sep 24 2001

nonn

The numbers corresponding to 2515 and 4983 are probable primes. [Farideh Firoozbakht, Oct 15 2009]

a(28) > 10000. - Giovanni Resta, Mar 16 2014

			n=5: 5! = 120 and prime(5) = 11, 120+11 = 131.
6 is listed because 6!+prime(6) = 720+13 = 733 is prime.

Cf. A063499 (Primes of form prime(n) + n!). [Alexander R. Povolotsky, Aug 13 2008]

[n: n in [1..200] | IsPrime(Factorial(n)+ NthPrime(n))]; // Vincenzo Librandi, Mar 05 2015

Do[ If[ PrimeQ[ n! + Prime[ n ] ], Print[ n ] ], {n, 1, 700} ]
Select[Range[1000], PrimeQ[#! + Prime[#]] &] (* Vincenzo Librandi, Mar 05 2015 *)

for(n=1, 100, if (isprime(n!+prime(n)), print1(n, ", ")))

More terms from Robert G. Wilson v, Sep 28 2001
More terms from John Sillcox (JMS21187(AT)aol.com), Apr 05 2003
a(25)-a(26) from Farideh Firoozbakht, Oct 15 2009
a(27) from Giovanni Resta, Mar 16 2014
a(28) from Michael S. Branicky, Sep 22 2024

A121926 a(n) = prime(n) + n!.

Original entry on oeis.org

3, 5, 11, 31, 131, 733, 5057, 40339, 362903, 3628829, 39916831, 479001637, 6227020841, 87178291243, 1307674368047, 20922789888053, 355687428096059, 6402373705728061, 121645100408832067, 2432902008176640071, 51090942171709440073, 1124000727777607680079

Offset: 1

N. J. A. Sloane, Aug 13 2008

nonn

It was conjectured by Alexander R. Povolotsky and proved (in SeqFan email exchange) by David L. Harden (with proof improvement from Daniel Berend) that no value of n exists such that ( n! + prime(n) ) yields an integral square. He also conjectured (see the link) that no value of n exists such that ( n! + prime(n) ) yields an integral m^k where k>1. - Alexander R. Povolotsky, Aug 13 2008

G. C. Greubel, Table of n, a(n) for n = 1..445
Carlos Rivera, Conjecture 59, The Prime Puzzles and Problems Connection.

Cf. A063499.

[NthPrime(n) + Factorial(n): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

Table[Prime[n]+n!,{n,30}] (* Harvey P. Dale, Mar 23 2012 *)

a(n)=prime(n) + n! \\ Charles R Greathouse IV, Jan 27 2015

Sum_{n>=1} 1/a(n) = 0.6657239270705138461513344444... - Alexander R. Povolotsky, Sep 22 2008

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}]

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

A064278 Numbers k such that k! + prime(k) is prime.

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A121926 a(n) = prime(n) + n!.

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

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