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

%I A122990 #17 Feb 16 2025 08:33:02
%S A122990 8,10,11,16,22,30,34,40,42,47,49,68,74,79,168,202,245,280,463,534,803,
%T A122990 936,958,1299,2455,2546,7391
%N A122990 Numbers m such that (1/99)*Sum_{k=1..m} k! = A007489(m)/99 is prime.
%C A122990 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, ...}.
%H A122990 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha132.htm">Factorizations of many number sequences</a>.
%H A122990 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>.
%t A122990 f=0;Do[f=f+n!;If[PrimeQ[f/99],Print[{n,f/99}]],{n,1,534}]
%t A122990 Position[Accumulate[Range[1000]!]/99,_?PrimeQ]//Flatten (* The program generates the first 23 terms of the sequence. *) (* _Harvey P. Dale_, Sep 21 2023 *)
%o A122990 (Python)
%o A122990 from math import factorial
%o A122990 from sympy import isprime, prime
%o A122990 def afind(limit, startk=8):
%o A122990     if startk <= 8 <= limit: print(8, end=", ")
%o A122990     f, s, startk = 1, 0, max(startk, 10)
%o A122990     for i in range(1, startk):
%o A122990         f *= i
%o A122990         s += f
%o A122990     for k in range(startk, limit+1):
%o A122990         f *= k
%o A122990         s += f
%o A122990         if isprime(s//99):
%o A122990             print(k, end=", ")
%o A122990 afind(535) # _Michael S. Branicky_, Jan 17 2022
%Y A122990 Cf. A007489, A003422 (Left factorial).
%K A122990 hard,more,nonn
%O A122990 1,1
%A A122990 _Alexander Adamchuk_, Oct 28 2006
%E A122990 a(21)-a(26) from _Michael S. Branicky_, Jan 17 2022
%E A122990 a(27) from _Michael S. Branicky_, Apr 05 2023