A125163 -id:A125163 - cp's OEIS frontend

A125162 a(n) is the number of primes of the form k! + n, 1 <= k <= n.

1, 1, 1, 1, 3, 1, 4, 0, 1, 1, 5, 1, 3, 0, 1, 1, 6, 1, 7, 0, 1, 1, 6, 0, 1, 0, 1, 1, 6, 1, 9, 0, 0, 0, 3, 1, 11, 0, 1, 1, 9, 1, 5, 0, 1, 1, 10, 0, 2, 0, 1, 1, 9, 0, 2, 0, 1, 1, 10, 1, 9, 0, 0, 0, 3, 1, 9, 0, 1, 1, 8, 1, 9, 0, 0, 0, 5, 1, 9, 0, 1, 1, 11, 0, 1, 0, 1, 1, 8, 0, 3, 0, 0, 0, 2, 1, 10, 0, 1, 1, 10, 1

Offset: 1

Alexander Adamchuk, Nov 21 2006

nonn

Note the triples of consecutive zeros in a(n) for n = {{32,33,34}, {62,63,64}, {74,75,76}, {92,93,94}, {116,117,118}, {122,123,124}, {140,141,142}, {152,153,154}, {158,159,160}, {182,183,184}, {200,201,202}, {206,207,208}, {212,213,214}, {218,219,220}, {242,243,244}, {272,273,274}, {284,285,286}, ...}. The middle index of most zero triples is a multiple of 3. See A125164.
The first consecutive quintuple of zeros has indices n = {294,295,296,297,298}, where the odd zero index n = 295 is not a multiple of 3.
Also for n >= 2, a(n) is the number of primes of the form k! + n for all k, since n divides k! + n for k >= n. Note that it is not known whether there are infinitely many primes of the form k! + 1; see A088332 for such primes and A002981 for the indices k. - Jianing Song, Jul 28 2018

			a(n) is the length of n-th row in the table of numbers k such that k! + n is a prime, 1 <= k <= n.
   n:  numbers k
   -------------
   1:  {1},
   2:  {1},
   3:  {2},
   4:  {1},
   5:  {2, 3, 4},
Thus a(1)-a(4) = 1, a(5) = 3.
See Example table link for more rows.

Michel Marcus, Example table

Cf. A125163 (indices of 0), A125164 (triples).

Table[Length[Select[Range[n],PrimeQ[ #!+n]&]],{n,1,300}]

a(n)=c=0;for(k=1,n,if(ispseudoprime(k!+n),c++));c
vector(100,n,a(n)) \\ Derek Orr, Oct 15 2014

Name clarified by Jianing Song, Jul 28 2018

Edited by Michel Marcus, Jul 29 2018

A125164 Positive integers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n) or (k! + 3n + 1) for any k.

Original entry on oeis.org

11, 21, 25, 31, 39, 41, 47, 51, 53, 61, 67, 69, 71, 73, 81, 91, 95, 99, 101, 107, 109, 111, 113, 121, 123, 125, 131, 135, 137, 141, 145, 151, 157, 161, 165, 171, 175, 177, 179, 181, 183, 191, 193, 201, 203, 207, 209, 211, 221, 223, 229, 231, 235, 237, 241, 243, 245, 249, 251, 255, 259

Offset: 1

Alexander Adamchuk, Nov 22 2006

nonn

3*a(n) is an index of the middle term in a triple of consecutive zeros in A125162. The indices of zeros in A125162 are listed in A125163.

Primes in this sequence form A115058.

			Triplets of consecutive terms in A125163: {32,33,34}, {62,63,64}, {74,75,76}, {92,93,94}, {116,117,118}, {122,123,124}, {140,141,142}, {152,153,154}, {158,159,160}, {182,183,184}, {200,201,202}, {206,207,208}, {212,213,214}, {218,219,220}, {242,243,244}, {272,273,274}, {284,285,286}.

Cf. A125162, A125163, A115058.

Select[Range[2,300],NoneTrue[Flatten[Table[k!+3*#+{-1,0,1},{k,#-1}]],PrimeQ]&] (* Harvey P. Dale, Jun 24 2025 *)

Edited and extended by Max Alekseyev, Feb 06 2010

A125211 a(n) = total number of primes of the form |k! - n|.

Original entry on oeis.org

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

Offset: 1

Alexander Adamchuk, Nov 23 2006

nonn

Numbers n such that a(n) = 0 are listed in A125212(n) = {1,2,10,16,28,34,36,40,46,50,51,52,56,57,58,64,66,70,76,78,82,86,87,88,92,93,94,96,100,...} Numbers n such that no prime exists of the form k! - n. Note the triples of consecutive zeros in a(n) for n = {{50,51,52}, {56,57,58}, {86,87,88}, {92,93,94}, ...}. Most zeros in a(n) have even indices. The middle index of most consecutive zero triples is odd and is a multiple of 3. Numbers n such that no prime exists of the form (k! - 3n - 1), (k! - 3n), (k! - 3n + 1) are listed in A125213(n) = {17,19,29,31,45,49,57,59,63,69,73,79,83,85,87,89,97,99,...}. The first pair of odd middle indices of zero triples that are not divisible by 3 is n = 325 and n = 329. They belong to the first septuplet of consecutive zeros in a(n): a(324)-a(330) = 0.

			a(4) = 3 because there are 3 primes of the form |k! - 4|:
1! - 4 = -3, 2! - 4 = -2, 3! - 4 = 2.
k! - 4 is composite for all k>3 because it is divisible by 4.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A125162 = number of primes of the form k! + n. Cf. A125163 = numbers n such that no prime exists of the form k! + n. Cf. A125164 = numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1). Cf. A125212, A125213.

Table[Length[Select[Range[n],PrimeQ[ #!-n]&]],{n,1,300}]

A125212 Numbers m such that no prime exists of the form |k! - m|; or A125211(m) = 0.

Original entry on oeis.org

1, 2, 10, 16, 28, 34, 36, 40, 46, 50, 51, 52, 56, 57, 58, 64, 66, 70, 76, 78, 82, 86, 87, 88, 92, 93, 94, 96, 100, 106, 112, 116, 120, 124, 126, 130, 134, 135, 136, 142, 144, 146, 147, 148, 154, 156, 160, 162, 166, 170, 171, 172, 176, 177, 178, 184, 186, 188, 189

Offset: 1

Alexander Adamchuk, Nov 23 2006

nonn

Note the triples of consecutive zeros in A125211 for n = {{50,51,52}, {56,57,58}, {86,87,88}, {92,93,94}, ...}. Most zeros in A125211 have even indices. The middle index of most consecutive zero triples in A125211 is odd and is a multiple of 3. Numbers n such that no prime exists of the form (k! - 3n - 1), (k! - 3n), (k! - 3n + 1) are listed in A125213. The first pair of odd middle indices of zero triples that are not divisible by 3 is n = 325 and n = 329. They belong to the first septuplet of consecutive zeros in A125211. A125211(n) = 0 for 7 consecutive terms from n = 324 to n = 330.

			A125211 begins {0,0,2,3,2,1,3,2,2,0,5,1,7,1,1,0,9,1,6,1,2,1,4,1,2,1,1,0,5,1,8,1,1,0,2,0,10,1,1,0,6,1,10,1,1,0,10,1,3,0,0,0,7,...}.
Thus a(1) = 1, a(2) = 2, a(3) = 10, a(10)-a(12) = {50,51,52}.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A125162 (number of primes of the form k! + n).
Cf. A125163 (numbers n such that no prime exists of the form k! + n).
Cf. A125164 (numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1)).
Cf. A125211, A125213.

k={};Do[If[Length[Select[Range[m],PrimeQ[#!-m]&]]==0,AppendTo[k,m]],{m,189}];k (* James C. McMahon, Dec 16 2024 *)

isok(m)={for(k=1, m-1, if(isprime(abs(k!-m)), return(0))); 1} \\ Andrew Howroyd, Dec 16 2024

A125213 Integers m such that no prime exists of the form abs(k! - 3m - 1), abs(k! - 3m), or abs(k! - 3m + 1).

Original entry on oeis.org

17, 19, 29, 31, 45, 49, 57, 59, 63, 69, 73, 79, 83, 85, 87, 89, 97, 99, 101, 107, 109, 115, 119, 121, 129, 131, 135, 139, 143, 149, 151, 157, 159, 161, 165, 169, 171, 173, 177, 179, 185, 187, 189, 195, 197, 199, 209, 213, 217, 219, 223, 227, 229, 233, 235, 249, 255, 259

Offset: 1

Alexander Adamchuk, Nov 23 2006

nonn

3*a(n) are the indices of the middle terms in the triples of consecutive zeros in A125211. The middle index of most consecutive zero triples in A125211 is odd and is a multiple of 3.

			Triplets of consecutive zeros in A125211: {50,51,52}, {56,57,58}, {86,87,88}, {92,93,94}, ...

Cf. A125162, A125163, A125164, A125211, A125212.

Edited and extended by Max Alekseyev, Feb 10 2010

A239321 Numbers n such that n - k! is never prime; or A175940(n) = 0.

Original entry on oeis.org

1, 2, 10, 16, 22, 28, 34, 36, 40, 46, 50, 51, 52, 56, 57, 58, 64, 66, 70, 76, 78, 82, 86, 87, 88, 92, 93, 94, 96, 100, 101, 106, 112, 116, 117, 118, 120, 124, 126, 130, 134, 135, 136, 142, 144, 146, 147, 148, 154, 156, 160, 162, 166, 170, 171, 172, 176, 177

Offset: 1

Derek Orr, Mar 15 2014

nonn

			51 - 0! = 51 - 1! = 50 is not prime. 51 - 2! = 49 is not prime. 51 - 3! = 45 is not prime. 51 - 4! = 27 is not prime. For k >= 5, 51 - k! is negative and thus not prime. Hence 51 is a member of this sequence since 51 - k! is not prime for any k.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A175940, A125163.

isok(n) = {k = 0; while (((nmk =(n - k!)) > 0), if (isprime(nmk), return (0)); k++;); return (1);} \\ Michel Marcus, Mar 16 2014

import sympy
from sympy import isprime
import math
def Prf(x):
  count = 0
  for i in range(x):
    if isprime(x-math.factorial(i)):
      count += 1
  return count
x = 1
while x < 10**3:
  if Prf(x) == 0:
    print(x)
  x += 1

A242040 Numbers n such that n + k! and n - k! are both prime for some k.

Original entry on oeis.org

4, 5, 6, 9, 11, 12, 13, 15, 17, 18, 21, 23, 25, 29, 30, 35, 37, 39, 42, 43, 45, 47, 53, 55, 60, 65, 67, 69, 72, 73, 77, 81, 83, 85, 95, 99, 102, 103, 105, 107, 108, 111, 113, 125, 127, 129, 131, 133, 137, 138, 143, 145, 149, 150, 151, 155, 157, 161, 163, 165, 173, 175, 180, 185, 187

Offset: 1

Derek Orr, Aug 12 2014

nonn

Subsequence of the complement of (A239321 union A125163).

			5 + 2! = 7 and 5 - 2! = 3 are both prime. Thus 5 is a member of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A239321, A125163, A241425, A245716.

a(n)=for(k=1,n,if(ispseudoprime(n+k!)&&ispseudoprime(n-k!),return(k)))
n=1;while(n<500,if(a(n),print1(n,", "));n++)

cp's OEIS Frontend

A125162 a(n) is the number of primes of the form k! + n, 1 <= k <= n.

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A125164 Positive integers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n) or (k! + 3n + 1) for any k.

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A125211 a(n) = total number of primes of the form |k! - n|.

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A125212 Numbers m such that no prime exists of the form |k! - m|; or A125211(m) = 0.

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A125213 Integers m such that no prime exists of the form abs(k! - 3m - 1), abs(k! - 3m), or abs(k! - 3m + 1).

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A239321 Numbers n such that n - k! is never prime; or A175940(n) = 0.

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A242040 Numbers n such that n + k! and n - k! are both prime for some k.

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