A380450 Number of integers k such that prime(n) - primorial(k) is prime.

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 1, 3, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 2, 0, 2, 2, 2, 1, 2, 2, 4, 2, 2, 3, 1, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 4, 2, 0, 3, 2, 2, 1, 2, 2, 2, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 0, 2, 3

Offset: 1

Author

Daniel D Gibson, Jun 22 2025

Keywords

• nonn

Comments

Conjecture A: Each value occurs an infinite number of times in the sequence.

Conjecture B: All natural numbers occur in the sequence.

Examples

			For prime(n=6): 13 - 2 = 11, and 13 - 6 = 7, so a(6) = 2.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A385210, A000040, A002110, A175974 (zeros (primes)), A115785 (record positions (primes)).

Programs

Mathematica

a[n_]:=Module[{c=0},Do[d=Prime[n]-Fold[Times, 1, Prime[Range[k-1]]];If[PrimeQ[d]&&d>0,c++],{k,n}];c];Array[a,90] (* James C. McMahon, Jun 27 2025 *)

PARI

pri(n) = vecprod(primes(n)); \\ A002110
a(n) = my(nb=0, p=prime(n)); for (k=0, n, if (isprime(p-pri(k)), nb++); ); nb; \\ Michel Marcus, Jun 22 2025

Extensions

More terms from Michel Marcus, Jun 22 2025