A104225 Decimal expansion of -x, where x is the real root of f(x) = 1 + Sum_{n} (twin_prime(n))*x^n.
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
Examples
-0.665070048764852292...
References
- 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.
Links
- 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.
Programs
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Mathematica
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 *)
Formula
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.
Extensions
Offset corrected by Sean A. Irvine, May 24 2025
a(99) corrected by Michael De Vlieger, May 24 2025