A393649 - OEIS

%I #5 Feb 24 2026 08:19:18

%S 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,

%T 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,

%U 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

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

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

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

%F 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))).

%e 0.621597313074143053465039519180428752081793741835412...

%t $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]]]

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

%Y Cf. A013661, A036966.

%Y 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).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Feb 24 2026