A122990 Numbers m such that (1/99)*Sum_{k=1..m} k! = A007489(m)/99 is prime.

8, 10, 11, 16, 22, 30, 34, 40, 42, 47, 49, 68, 74, 79, 168, 202, 245, 280, 463, 534, 803, 936, 958, 1299, 2455, 2546, 7391

Offset: 1

Author

Alexander Adamchuk, Oct 28 2006

Keywords

• hard
• more
• nonn

Comments

A007489(n) = Sum_{k=1..m} k! = (!(n+1) - 1) = A003422(n+1) - 1 = {0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, ...}. A007489(n) is divisible by 99 for n=8 and n>9. Corresponding primes of the form (!(n+1) - 1)/99 are {467, 40787, 443987, 225498914387, 11895484822660898387, 2771826449193354891007108898387, 3072603482270933019578343003268898387, ...}.

Links

• Hisanori Mishima, Factorizations of many number sequences.
• Eric Weisstein's World of Mathematics, Left Factorial.

Crossrefs

Cf. A007489, A003422 (Left factorial).

Programs

Mathematica

f=0;Do[f=f+n!;If[PrimeQ[f/99],Print[{n,f/99}]],{n,1,534}]
Position[Accumulate[Range[1000]!]/99,?PrimeQ]//Flatten (* The program generates the first 23 terms of the sequence. *) (* _Harvey P. Dale, Sep 21 2023 *)

Python

from math import factorial
from sympy import isprime, prime
def afind(limit, startk=8):
    if startk <= 8 <= limit: print(8, end=", ")
    f, s, startk = 1, 0, max(startk, 10)
    for i in range(1, startk):
        f *= i
        s += f
    for k in range(startk, limit+1):
        f *= k
        s += f
        if isprime(s//99):
            print(k, end=", ")
afind(535) # Michael S. Branicky, Jan 17 2022

Extensions

a(21)-a(26) from Michael S. Branicky, Jan 17 2022

a(27) from Michael S. Branicky, Apr 05 2023