A064164 EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).

8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Author

R. K. Guy, Sep 20 2001

Keywords

• nonn
• nice

Comments

The complement of this sequence (A064295) is a superset of A002981, that is, terms of A002981 do not appear in this sequence.

Hardy & Subbarao prove that this sequence is infinite, see their Theorem 2.12. - Charles R Greathouse IV, Sep 10 2015

Links

• Amiram Eldar, Table of n, a(n) for n = 1..120
• G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109:6 (2002), pp. 554-559.
• H. Mishima, Factors of N!+1

Crossrefs

The smallest associated primes p are given in A064229.

Cf. A002981, A064295.

Programs

Mathematica

Do[k = 1; While[p = Prime[k]; k < 10^8 && Not[ Nor[ Mod[n! + 1, p] != 0, Mod[p, n] == 1]], k++ ]; If[k != 10^8, Print[n, " ", p]], {n, 2, 88}]

PARI

is(n)=my(f=factor(n!+1)[,1]); for(i=1,#f, if(f[i]%n != 1, return(n>1))); 0 \\ Charles R Greathouse IV, Sep 10 2015

Extensions

Corrected and extended by Don Reble, Sep 23 2001