A115785 -id:A115785 - cp's OEIS frontend

A115786 Smallest prime number p such that p + 2#, p + 3#, ..., p + prime(n)# are all prime, where x# = A034386(x) is the primorial.

3, 5, 11, 17, 41, 41, 41, 41, 86351, 86351, 235313357, 729457511, 99445156397, 818113387907, 7986903815771, 29065965967667

Offset: 1

Rick L. Shepherd, Jan 31 2006

Subset of A001359 (lesser of twin primes).

From a(2) = 5 on, also a subset of A022004: first element of prime triples (p, p+2, p+6).-- It could make sense to add a(0) = 2, the smallest prime (with empty further restriction "p + prime(n)# prime for 1 <= n <= 0"). - M. F. Hasler, Apr 29 2015

			a(11) = 235313357 because 235313357, 235313357 + 2, 235313357 + 2*3, 235313357 + 2*3*5, ... and 235313357 + 2*3*5*7*11*13*17*19*23*29*31 are all prime and there is no smaller prime with this property.

C. Rivera, Puzzle 782: Prime-Generating non-polynomials, primepuzzles.net, April 4, 2015
C. Rivera, Puzzle 350. Primes & primorials, primepuzzles.net, 2006

Cf. A001359, A002110, A115785 (for p - p(i)#).

a(12) from Don Reble, Feb 15 2006

More terms from Jens Kruse Andersen, Feb 28 2006

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

Original entry on oeis.org

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

Daniel D Gibson, Jun 22 2025

nonn

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

Conjecture B: All natural numbers occur in the sequence.

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

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

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

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 *)

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

More terms from Michel Marcus, Jun 22 2025

cp's OEIS Frontend

A115786 Smallest prime number p such that p + 2#, p + 3#, ..., p + prime(n)# are all prime, where x# = A034386(x) is the primorial.

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