A390839 a(n) = Product_{k=1..n} (prime(k) + 1)/A001223(k), where A001223 lists the prime gaps, rounded to the nearest integer if necessary.

1, 3, 6, 18, 36, 216, 756, 6804, 34020, 136080, 2041200, 10886400, 103420800, 2171836800, 23890204800, 191121638400, 1720094745600, 51602842368000, 533229371136000, 9064899309312000, 326336375135232000, 4024815293334528000, 80496305866690560000, 1126948282133667840000, 12678168174003763200000

Offset

0,2

Comments

The product is actually conjectured to be an integer for all n. Can anyone prove it?

Sequences A390840, A390841, A390842, A390843 and A390844 are related to this question.

Links

Paolo Xausa, Table of n, a(n) for n = 0..448 (terms 0..99 from M. F. Hasler)

M. F. Hasler, in reply to Sebastian M. Ruiz, All integers?, PrimenumbersTheory group.io, Nov. 21, 2025.

Maple

a:= proc(n) option remember; `if`(n<1, 1, (p->
      a(n-1)*(1+p(n))/(p(n+1)-p(n)))(ithprime))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 22 2026

Mathematica

FoldList[Times, 1, Map[(#[[1]] + 1)/(#[[2]] - #[[1]]) &, Partition[Prime[Range[25]], 2, 1]]] (* Paolo Xausa, Apr 22 2026 *)

Prog

(PARI) A390839(n, p=2)=prod(k=1, n, (p+1)/(-p+p=prime(k+1))) \\ M. F. Hasler, Nov 21 2025

Crossrefs

Cf. A080082 (where gap 2*prime(k) occurs first), A001223 (prime gaps).

Cf. A390840 (number of primes <= A080082(n) such that prime(n) | q+1).

Cf. A390841, A390842, A390843, A390844 (2-, 3-, 5- and 7-valuation of the product).

Cf. A054640.

Sequence in context: A268529 A026532 A160505 * A081150 A362014 A216813

Adjacent sequences: A390836 A390837 A390838 * A390840 A390841 A390842

Keyword

nonn,less

Author

M. F. Hasler, Nov 21 2025

Status

approved