A074269 -id:A074269 - cp's OEIS frontend

A072508 Decimal expansion of Backhouse constant.

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

The reciprocal (A088751) of Backhouse's constant is the real zero of a certain power series. - T. D. Noe, Oct 14 2003

			1.4560749485826896713995953...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.5, p. 294.

S. R. Finch, Backhouse's constant, 1995. [Cached copy, with permission]
Philippe Flajolet, in response to the previous document from S. R. Finch, Backhouse's constant, 1995.
S. R. Finch, Kalmar's Composition Constant, Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003. [Cached copy, with permission]
S. R. Finch, Kalmar's composition constant, June 5, 2003. [A different version. Cached copy, with permission of the author]
Simon Plouffe, The Backhouse constant.
Eric Weisstein's World of Mathematics, Backhouse's Constant.

Continued fraction is in A074269.

Cf. A030010, A088751, A247818.

RealDigits[-1/x /. FindRoot[0 == 1 + Sum[x^n Prime[n], {n, 1000}], {x, {0, 1}}, WorkingPrecision -> 100]][[1]] (* T. D. Noe, corrected Apr 26 2013 *)

A104225 Decimal expansion of -x, where x is the real root of f(x) = 1 + Sum_{n} (twin_prime(n))*x^n.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Apr 01 2005

			-0.665070048764852292...

S. R. Finch, "Kalmar's Composition Constant", Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003.
Martin Gardner, "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

S. R. Finch, Backhouse's constant. 1995 [Cached copy, with permission]
Philippe Flajolet, in response to the previous document from S. R. Finch, Backhouse's constant, 1995
S. R. Finch, Kalmar's Composition Constant, Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003. [Cached copy, with permission]
Eric Weisstein's World of Mathematics, Backhouse's Constant.
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A072508, A074269, A077800, A088751.

ps={}; Do[If[PrimeQ[n]&&PrimeQ[n+2], AppendTo[ps, {n, n+2}]], {n, 3, 40001, 2}];
ps=Flatten[ps];
RealDigits[ -x /. FindRoot[0==1+Sum[x^n ps[[n]], {n, 1000}], {x, -0.665}, WorkingPrecision->100]][[1]] (* T. D. Noe *)

Decimal expansion of -x where x is the real root of f(x) = 1 + 3x + 5x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + 17x^7 + 19x^8 + 29x^9 + 31x^10 + 41x^11 + 43x^12 + 59x^13 + 61x^14 + 71x^15 + 73x^16 + ... where for n>0 the coefficient of x^n is the n-th twin prime.

Offset corrected by Sean A. Irvine, May 24 2025

a(99) corrected by Michael De Vlieger, May 24 2025

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A072508 Decimal expansion of Backhouse constant.

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