A371321 Decimal expansion of Sum_{k>=0} 1/A007018(k).

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

Offset: 1

Author

Amiram Eldar, Mar 19 2024

Keywords

• nonn
• cons

Comments

The corresponding alternating sum, Sum_{k>=0} (-1)^k/A007018(k), equals Cahen's constant (A118227).

Duverney et al. (2018) proved that this constant is transcendental.

Called the "Kellogg-Curtiss constant" by Sondow (2021), after the American mathematicians Oliver Dimon Kellogg (1878-1932) and David Raymond Curtiss (1878-1953).

The Engel expansion of this constant is 1 followed by the Sylvester sequence (A000058, see the Formula section).

Examples

			1.69103020675725397443566284314574179380857724257952...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7, p. 436.

Links

• Brenda S. Baker and Edward G. Coffman, Jr., A tight asymptotic bound for next-fit-decreasing bin-packing, SIAM Journal on Algebraic Discrete Methods, Vol. 2, No. 2 (1981), pp. 147-152.
• Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa, Transcendence of numbers related with Cahen's constant, Moscow Journal of Combinatorics and Number Theory, Vol. 8, No. 1 (2018), pp. 57-69; alternative link.
• Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa. Irrationality exponents of certain fast converging series of rational numbers, Tsukuba Journal of Mathematics, Vol. 44, No. 2 (2020), pp. 235-250; alternative link.
• Chan C. Lee and Der-Tsai Lee, A simple on-line bin-packing algorithm, Journal of the ACM (JACM), Vol. 32, No. 3 (1985), pp. 562-572. See p. 566.
• Iekata Shiokawa, Irrationality exponents of certain alternating serries, Analytic Number Theory and Related Topics, Vol. 2162 (2020), pp. 210-215.
• Jonathan Sondow, Irrationality and Transcendence of Alternating Series via Continued Fractions, in: A. Bostan and K. Raschel (eds.), Transcendence in Algebra, Combinatorics, Geometry and Number Theory, TRANS 2019. Springer Proceedings in Mathematics & Statistics, Vol. 373, Springer, Cham, 2021; arXiv preprint, arXiv:2009.14644 [math.NT], 2020.
• Andrew Twigg and Eduardo C. Xavier, Locality-preserving allocations problems and coloured bin packing, Theoretical Computer Science, Vol. 596 (2015), pp. 12-22.
• Wikipedia, Bin packing problem.
• Wikipedia, Harmonic bin packing.
• Wikipedia, Next-fit-decreasing bin packing.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000058, A007018, A118227, A242724.

Programs

Mathematica

s[0] = 2; s[n_] := s[n] = s[n - 1]^2 - s[n - 1] + 1; kmax = 1; FixedPoint[RealDigits[Sum[1/(s[k] - 1), {k, 0, kmax += 10}], 10, 120][[1]] &, kmax] (* after Jean-François Alcover at A118227 *)

PARI

c = 1; 1 + suminf(k = 1, c += c^2; 1/c) \\ after Charles R Greathouse IV at A118227

Formula

Equals 1 + Sum_{k>=1} 1/(Product_{i=0..k-1} A000058(i)).