A394630 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, 3, 4, 5, 7, 25, 175, 3848, 461, 4174, 68860, 3773, 218926, 354713, 2910933, 30498, 9788597, 21959114, 938903578, 294933055, 910526399, 20426640940, 114303248603, 434849901116, 79915701398721, 3449315100163, 212301237060282, 927957085532368

Offset

0,5

Comments

Upon the observation that 5^2 + 4! = 25 + 24 = 49 = 7^2, it is natural to ask whether there are primes p, q such that q^2 - p^2 = n! for all n >= 4. This sequence gives the index (or "rank", or "primepi" values) of the corresponding primes p.

Links

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

Formula

a(n) = A000720(A394631(n)) for all n. - M. F. Hasler, Apr 05 2026

Example

For 0! = 1, there are no two primes p, q such that p^2 + 1 = q^2, therefore a(0) = 0.
Similarly, for n = 1, 2 and 3, with n! = 1, 2 and 6, there are no two primes p, q such that p^2 + n! = q^2, therefore a(n) = 0 for n = 1, 2 and 3.
For n = 4, we have the primes p = 5 and q = 7 such that 5^2 + 4! = 25 + 24 = 49 = 7^2, so a(4) = primepi(5) = 3.
For n = 5, we have 7^2 + 5! = 13^2, so a(5) = primepi(7) = 4.
For n = 6, we have 11^2 + 6! = 29^2, so a(6) = primepi(11) = 5.

Prog

(PARI) apply( {A394630(n, f=n!)=n>3&&forprime(p=3, oo, issquare(p^2+f, &q)&& isprime(q)&& return(primepi(p)))}, [0..15]) \\ For illustration; better use {A394630(n)=primepi(A394631(n))} with more efficient code given there.
(Sidef) sidef -N 'USE_PRIMECOUNT=1' -E 'for n in (0..30) { n!.diff_of_squares.first_by{.all_prime} || [0, 0] -> last.primepi.print(", ") }' # Daniel Suteu, Apr 09 2026

Crossrefs

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

See A394631 for the primes p = prime(a(n)), A394632 for the primes q = sqrt(p^2 + n!).

See A394629 for the total number of solutions (p, q): n! = q^2 - p^2.

Sequence in context: A035359 A392634 A378963 * A325410 A269719 A214626

Adjacent sequences: A394627 A394628 A394629 * A394631 A394632 A394633

Keyword

nonn

Author

M. F. Hasler, Apr 04 2026

Extensions

More terms from Daniel Suteu, Apr 09 2026

Status

approved