A390841 2-valuation of Product_{k=1..n} (prime(k) + 1) / (prime(k+1) - prime(k)).

0, 0, 1, 1, 2, 3, 2, 2, 2, 4, 4, 8, 7, 7, 7, 10, 10, 11, 11, 11, 13, 13, 15, 16, 14, 13, 13, 14, 15, 14, 14, 19, 20, 20, 21, 21, 23, 23, 23, 25, 25, 26, 26, 31, 30, 30, 31, 31, 34, 35, 34, 34, 37, 37, 38, 38, 40, 40, 43, 42, 42, 43, 43, 43, 45, 44, 44, 45, 45

Offset

0,5

Comments

The given product is likely to be an integer for all n, because the cumulated numerators are extremely likely to have more of any given prime factor than the smallish denominators which are the prime gaps. But can it be proved that this is indeed always the case for all prime factors?

The sequence appears to grow slightly sub-linearly: a(10^3) = 460, a(10^4) = 3996, a(10^5) = 35962.

Links

M. F. Hasler, Table of n, a(n) for n = 0..10000, Nov 21 2025

Formula

a(n) = A007814(A390839(n)). - Paolo Xausa, Apr 22 2026

Example

The first factors of the product are (2+1)/1 = 3, (3+1)/2 = 2, (5+1)/2 = 3, (7+1)/4 = 2, (11+1)/2 = 6, (13+1)/4 = 7/2, (17+1)/2 = 9, (19+1)/4 = 5, (23+1)/6 = 4, (29+1)/2 = 15, (31+1)/6 = 16/3, ...
Therefore, for the product of the first n = 0, 1, 2, 3, ... of these fractions, the cumulated number of powers of 2 is 0, 0, 1, 1, 2, 3, 2, 2, 2, 4, 4, 8, ...

Maple

b:= proc(n) option remember; `if`(n<1, 1, (p->
      b(n-1)*(1+p(n))/(p(n+1)-p(n)))(ithprime))
    end:
a:= n-> padic[ordp](b(n), 2):
seq(a(n), n=0..68);  # Alois P. Heinz, Apr 22 2026

Mathematica

IntegerExponent[FoldList[Times, 1, Map[(#[[1]] + 1)/(#[[2]] - #[[1]]) &, Partition[Prime[Range[100]], 2, 1]]], 2] (* Paolo Xausa, Apr 22 2026 *)

Prog

(PARI) concat(s=0, vector(99, i, s+=valuation((prime(i)+1)/(prime(i+1)-prime(i)), 2)))

Crossrefs

Cf. A007814 (2-valuation), A023512 (2-valuation of prime(n) + 1).

Cf. A390842, A390843, A390844 (similar for 3-, 5- and 7-valuation).

Cf. A390839.

Sequence in context: A187757 A286529 A306225 * A373249 A077199 A145390

Adjacent sequences: A390838 A390839 A390840 * A390842 A390843 A390844

Keyword

nonn

Author

M. F. Hasler, Nov 21 2025

Status

approved