A064152 Erdős primes: primes p such that all p-k! for 1 <= k! < p are composite.

2, 101, 211, 367, 409, 419, 461, 557, 673, 709, 769, 937, 967, 1009, 1201, 1259, 1709, 1831, 1889, 2141, 2221, 2309, 2351, 2411, 2437, 2539, 2647, 2837, 2879, 3011, 3019, 3041, 3049, 3079, 3163, 3217, 3221, 3359, 3389, 3499, 3593, 3671, 3709, 3833, 3851

Offset: 1

Author

Felice Russo, Sep 13 2001

Keywords

• easy
• nonn

Comments

Numbers of Erdős primes <= 10^j for j = 1,2,3, ... are 1, 1, 13, 95, 901, 7875, 71140, 646242, 5901409, ... For large j the asymptotic law seems to be #E(10^j) ~ (1/8)*(10^j/(j*log(10))). If so the sequence is infinite.

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A2, p. 11.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7875 from T. D. Noe)

Crossrefs

Cf. A000142, A065381.

Programs

Mathematica

q[n_] := Module[{k = 1}, While[k! < n && ! PrimeQ[n - k!], k++]; k! >= n]; Select[Prime[Range[550]], q] (* Amiram Eldar, Mar 21 2024 *)

PARI

{ n=0; for (m=1, 10^9, p=prime(m); k=f=b=1; while ((f*=k) < p, if (isprime(p-f), b=0; break); k++); if (b, write("b064152.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 09 2009