A393649 Decimal expansion of Product_{p prime} (1 - 1/p^2 + 1/p^6).

6, 2, 1, 5, 9, 7, 3, 1, 3, 0, 7, 4, 1, 4, 3, 0, 5, 3, 4, 6, 5, 0, 3, 9, 5, 1, 9, 1, 8, 0, 4, 2, 8, 7, 5, 2, 0, 8, 1, 7, 9, 3, 7, 4, 1, 8, 3, 5, 4, 1, 2, 9, 4, 8, 5, 8, 9, 7, 7, 4, 4, 5, 3, 1, 7, 2, 4, 0, 7, 8, 9, 5, 2, 6, 6, 0, 4, 4, 6, 5, 1, 1, 9, 8, 1, 7, 8, 8, 2, 5, 7, 0, 8, 4, 0, 2, 8, 1, 5, 7, 9, 2, 1, 3, 8

Offset

0,1

Comments

The asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a cubefull number (A036966).

Links

Table of n, a(n) for n=0..104.

Formula

Equals (1/zeta(2)) * Product_{p prime} (1 + 1/(p^4*(p^2-1))).

In general, the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a k-full number (a number whose prime factorization has no exponent that is smaller than k) is (1/zeta(2)) * Product_{p prime} (1 + 1/(p^(2*k-2)*(p^2-1))).

Example

0.621597313074143053465039519180428752081793741835412...

Mathematica

$MaxExtraPrecision = 1000; Module[{m = 1000, c}, c = LinearRecurrence[{0, 1, 0, 0, 0, -1}, {0, -2, 0, -2, 0, 4}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]]

Prog

(PARI) prodeulerrat(1 - 1/p^2 + 1/p^6)

Crossrefs

Cf. A013661, A036966.

The asymptotic probability that the greatest common divisor of two positive integers selected at random is: A010701 (not 5-rough), A010722 (5-rough), A020773 (even), A059956 (1), A082020/10 (2), A152627 (odd), A182448 (square), A185197 (even power of 2), A215267 (squarefree), A217739 (power of 2), A222056 (prime), A343359 (cubefree), A393646 (cube), A393647 (exponentially odd number), A393648 (powerful), this constant (cubefull), A393650 (3-smooth), A393651 (prime power), A393652 (perfect power).

Sequence in context: A224518 A375741 A363684 * A265986 A334942 A388786

Adjacent sequences: A393646 A393647 A393648 * A393650 A393651 A393652

Keyword

nonn,cons

Author

Amiram Eldar, Feb 24 2026

Status

approved