A084893 Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).

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

Offset: 3

Author

Eric W. Weisstein, Jun 10 2003

Keywords

• nonn
• cons

Comments

This constant, A_3, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_2 = A084892. - Amiram Eldar, Oct 16 2020

Examples

			118.6924619727642846261669938137118...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Links

• B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
• Eric Weisstein's World of Mathematics, Abelian Group.

Crossrefs

Cf. A000688, A021002, A063966, A084892.

Programs

Mathematica

m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/3]*Zeta[2/3]*Product[Zeta[j/3], {j, 4, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)

PARI

prodinf(k=1, if (k!=3, zeta(k/3), 1)) \\ Michel Marcus, Oct 16 2020