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

1,1

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)

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 *)

Prog

(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

Crossrefs

Cf. A000142, A065381.

Sequence in context: A195903 A042249 A157338 * A162353 A088272 A125819

Adjacent sequences: A064149 A064150 A064151 * A064153 A064154 A064155

Keyword

easy,nonn

Author

Felice Russo, Sep 13 2001

Status

approved