A175940 -id:A175940 - cp's OEIS frontend

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

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

A181676 Number of ways of writing n = m + f! with m a semiprime and f > 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 2, 2, 2, 0, 0, 2, 3, 1, 0, 0, 1, 1, 1, 2, 1, 0, 1, 3, 3, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 2, 0, 0, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 3, 2, 3, 1, 1, 1, 3, 2, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 3, 1, 0, 1, 2, 1, 1, 1, 2, 1, 0, 2, 2, 3, 2, 0, 1, 2, 3, 1, 2, 2, 2, 1, 1, 1, 2, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 04 2010

nonn

Number of partitions of n into a semiprime and a factorial.

			a(16)=3 because 16 = 2*5 + 3! = 2*7 + 2! = 3*5 + 1!.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000142, A001358, A175940.

N:= 200: # for a(1) .. a(N)
R:= Vector(N):
S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]):
F:= 1: v:= 1:
for i from 2 do v:= v*i; if v > N then break fi; F:= F,v od:
F:= [F]:
for s in S do
  for f in F do
    v:= s+f;
     if v > N then break fi;
     R[v]:= R[v]+1
od od:
convert(R,list); # Robert Israel, Sep 20 2024

A241424 Largest number k > 0 such that n - k! is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 1, 2, 2, 1, 2, 3, 3, 0, 3, 1, 3, 1, 2, 0, 3, 1, 3, 1, 2, 0, 3, 1, 3, 4, 4, 0, 4, 1, 4, 1, 2, 0, 4, 0, 4, 1, 2, 0, 4, 1, 4, 1, 2, 0, 4, 1, 3, 0, 0, 0, 4, 1, 4, 0, 0, 0, 3, 1, 4, 1, 2, 0, 4, 0, 4, 1, 2, 0, 4, 1, 3, 1, 2, 0, 4, 0, 3, 1, 2, 0, 4, 1, 4, 0, 0, 0, 3, 1, 4, 0, 0, 0, 4

Offset: 1

Derek Orr, Aug 08 2014

nonn

If k > n, n - k! is negative and therefore, not prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A175940, A245715.

a:= proc(n) local k, r;
r:= 0;
for k from 1 do
   if k! >= n then return r
   elif isprime(n-k!) then r:= k
   fi
od
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 10 2014

a[n_] := Module[{k, r = 0}, For[k = 1, True, k++, If[k! >= n, Return[r], If[PrimeQ[n - k!], r = k]]]];
Array[a, 100] (* Jean-François Alcover, Jul 27 2020, after Maple *)

a(n)=forstep(k=n,1,-1,if(ispseudoprime(n-k!),return(k)))
n=1;while(n<150,print1(a(n),", ");n++)

A242041 Number of k >= 1 that make n + k! and n - k! both prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Derek Orr, Aug 12 2014

nonn

0 <= a(n) <= min{A125162(n), A175940(n)}.

a(n) < A020639(n). - Robert Israel, Aug 12 2014

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

Cf. A020639, A125162, A175940.

a:= n ->  nops(select(k -> isprime(n+k!) and isprime(n-k!),
      [$1 ..min(numtheory:-factorset(n))-1])):
0, seq(a(n),n=2..100); # Robert Israel, Aug 12 2014

a(n)=c=0;for(k=1,n,if(ispseudoprime(n+k!)&&ispseudoprime(n-k!),c++));return(c)
vector(100,n,a(n))

cp's OEIS Frontend

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

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A181676 Number of ways of writing n = m + f! with m a semiprime and f > 0.

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Maple

A241424 Largest number k > 0 such that n - k! is prime, or 0 if no such k exists.

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PARI

A242041 Number of k >= 1 that make n + k! and n - k! both prime.

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Maple

PARI