A390842 3-valuation of Product_{k=1..n} (prime(k) + 1) / (prime(k+1) - prime(k)).

0, 1, 1, 2, 2, 3, 3, 5, 5, 5, 6, 5, 5, 6, 6, 6, 8, 9, 8, 8, 10, 9, 9, 9, 11, 11, 12, 12, 15, 15, 16, 16, 16, 17, 17, 18, 17, 16, 16, 16, 16, 18, 18, 19, 19, 21, 20, 19, 19, 20, 20, 21, 22, 22, 23, 23, 23, 26, 25, 25, 26, 26, 27, 27, 28, 28, 29, 28, 28, 29, 29, 29, 31, 30, 29, 29, 29, 30, 30, 31, 31, 32, 32, 35, 34, 34, 34, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 41, 43, 41

Offset

0,4

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) = 351, a(10^4) = 2864, a(10^5) = 25378.

Links

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

Formula

a(n) = A007949(A390839(n)). - Paolo Xausa, Apr 22 2026

Example

For n=0, we have the empty product equal to 1, with 3-valuation 0.
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 3 is 0, 1, 1, 2, 2, 3, 3, 5, 5, 5, 6, 5, ...

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), 3):
seq(a(n), n=0..99);  # Alois P. Heinz, Apr 22 2026

Mathematica

IntegerExponent[FoldList[Times, 1, Map[(#[[1]] + 1)/(#[[2]] - #[[1]]) &, Partition[Prime[Range[100]], 2, 1]]], 3] (* Paolo Xausa, Apr 22 2026 *)

Prog

(PARI) concat(s=0, vector(99, i, s+=valuation((prime(i)+1)/(prime(i+1)-prime(i)), 3)))

Crossrefs

Cf. A390841, A390843, A390844 (similar for 2-, 5- and 7-valuation).

Cf. A227990 (3-valuation of prime(n) + 1).

Cf. A007949, A390839.

Sequence in context: A072729 A057872 A059974 * A045767 A108221 A169682

Adjacent sequences: A390839 A390840 A390841 * A390843 A390844 A390845

Keyword

nonn

Author

M. F. Hasler, Nov 21 2025

Status

approved