A394633 Least index k such that prime(k)^2 - n! is again the square of a prime, or 0 if no such prime exists.

0, 0, 0, 0, 4, 6, 10, 21, 48, 197, 3854, 911, 4705, 69137, 25797, 232944, 466932, 3110924, 4670197, 20851430, 80081977, 987827086, 1470388038, 6545264485, 35778005903, 179227960118, 797794938348, 79971185783965, 17084369697743, 227738431773765, 1024021410267278

Offset

0,5

Comments

Upon the observation that 5^2 + 4! = 7^2 and 7^2 + 5! = 13^2, it is natural to ask whether for all n >= 4, there are primes p, q such that p^2 + n! = q^2.

This sequence gives the indices (or "primepi" values) of the primes q.

At present, it appears to be much easier to compute the primes q = A394632(n) than a(n) = primepi(q) for primes as huge as prime(a(30) ~ 10^15) and beyond.

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = A000720(A394632(n)).

Example

For n < 4 there are no primes p, q such that p^2 + n! = q^2, since 3^2 - 2^2 = 5 and 5^2 - 3^2 = 16 is already too large.
For n = 4 we have 5^2 + 4! = 7^2 and 7 is the 4th prime, so a(4) = 4.
For n = 5 we have 7^2 + 5! = 13^2 and 11 is the 6th prime, so a(5) = 6.
For n = 6 we have 11^2 + 6! = 29^2 and 29 is the 10th prime, so a(6) = 10.
See A394631 for more.

Prog

(PARI) apply( {A394633(n, f=n!)=n>3&&forprime(p=3, oo, issquare(p^2+f, &q)&& isprime(q)&& return(primepi(q)))}, [0..16])
/* or better: */  A394633(n) = primepi(A394632(n))

Crossrefs

Cf. A001248 (squares of primes), A000142 (factorial), A000720 (primepi).

See A394632 for prime(a(n)) and further information, A394629 for the number of solutions (p, q).

Sequence in context: A185913 A243119 A277343 * A077065 A131867 A252656

Adjacent sequences: A394630 A394631 A394632 * A394635 A394636 A394637

Keyword

nonn

Author

M. F. Hasler, Apr 04 2026

Extensions

More terms from Daniel Suteu, Apr 07 2026

Status

approved