A088751 -id:A088751 - cp's OEIS frontend

A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

-1, -3, 1, 1, -1, 5, -19, -9, 41, -103, 17, 289, -169, 331, -689, -4991, 3999, 7833, -6509, 21827, -22165, -87637, 119441, -190981, -152513, 1482023, -425985, -1045091, 1071237, -14108791, 5845271, 39852203, -35832801, 54451699, 44061359, -435442725, 261309855, -22217917

Offset: 1

Raul Prisacariu, Jan 08 2024

sign

Whittaker's root series formula is applied to 1 + Sum_{k>=1} prime(k) x^k. The following infinite series that converges to the negative inverse of Backhouse's constant (-x) is obtained:
x = -1/(1*2) - 3/(2*1) + 1/(1*1) + 1/(1*2) - 1/(2*3) + 5/(3*7) - 19/(7*10) - 9/(10*13) + 41/(13*21) - 103/(21*26) + 17/(26*33) + 289/(33*53) ...
The denominators of the infinite series are obtained by multiplying the absolute values of 2 consecutive terms from the sequence A030018.

			a(1) = -1;
a(2) = -3;
a(3) = -det ToeplitzMatrix((3,2),(3,5)) = 1;
a(4) = -det ToeplitzMatrix((3,2,1),(3,5,7)) = 1;
a(5) = -det ToeplitzMatrix((3,2,1,0),(3,5,7,11)) = -1;
a(6) = -det ToeplitzMatrix((3,2,1,0,0),(3,5,7,11,13)) = 5;
a(7) = -det ToeplitzMatrix((3,2,1,0,0,0),(3,5,7,11,13,17)) = -19.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A088751, A030018, A072508.

a(1) = -1.

For n > 1, a(n) = -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n))), where c(0)=1 and c(n) is the n-th prime number.

a(21)-a(38) from Stefano Spezia, Jan 09 2024

A030010 Inverse Euler transform of primes.

Original entry on oeis.org

2, 0, 1, 0, 2, -3, 2, -4, 4, -3, 4, -5, 10, -21, 20, -18, 34, -46, 64, -99, 126, -182, 258, -319, 464, -685, 936, -1352, 1888, -2570, 3690, -5188, 7292, -10501, 14742, -20766, 29610, -41650, 59052, -84338, 119602, -170279, 242256, -343356, 489550, -698073

Offset: 1

N. J. A. Sloane

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			(1-x)^(-2) * (1-x^3)^(-1) * (1-x^5)^(-2) * (1-x^6)^3 * (1-x^7)^(-2) * ... = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
N. J. A. Sloane, Transforms

Cf. A000040, A030011, A072508, A088751.

pp = Prime[Range[n = 100]]; s = {};
For[i = 1, i <= n, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i-d]], {d, i-1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, May 10 2019 *)

Product_{k>=1} 1/(1-x^k)^{a(k)} = 1 + Sum_{n>=1} prime(n) * x^n.
From Vaclav Kotesovec, Oct 09 2019: (Start)
a(n) ~ -(-1)^n * A072508^n / n.
a(n) ~ -(-1)^n / (n * A088751^n). (End)

A072508 Decimal expansion of Backhouse constant.

Original entry on oeis.org

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

A247818 Decimal expansion of 1/(theta*P'(theta)), a constant appearing in the asymptotic evaluation of the coefficients q_n in 1/(1+P(x)), where P(x) is the generating function of the primes and theta the unique zero of P(x) in [-3/4, 0].

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 24 2014

			-0.622306574570085664621341181270009605130784301479...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 294 and p. 551.

Cf. A030018, A072508, A088751.

digits = 100; P[x_] := 1 + Sum[Prime[n]*x^n, {n, 1, 1000}]; PPrime[x_] := Sum[n*Prime[n]*x^(n-1), {n, 1, 1000}]; theta = x /. FindRoot[P[x] == 0, {x, -3/4}, WorkingPrecision -> digits+5]; RealDigits[1/(theta*PPrime[theta]), 10, digits] // First

q_n ~ (1/(theta*P'(theta))) * (1/theta^n).

A114041 Decimal expansion of -x, the real root of the power series with semiprime coefficients.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Feb 01 2006

This is the semiprime analog of A088751.

Terms computed by T. D. Noe.

			-0.36986874348484794489584877...

Cf. A001358, A072508, A088751.

A001358:= Select[Range[3000], PrimeOmega[#] == 2 &]; RealDigits[-x/.FindRoot[-1 == Sum[A001358[[j]]*x^j, {j, 500}], {x, {0, 0.5}}, WorkingPrecision -> 105], 10, 100][[1]]//First (* G. C. Greubel, Dec 31 2019 *)

Digits of -x where x is the real root of 1 + 4x + 6x^2 + 9x^3 + 10x^4 + 14x^5 ... = 1 + Sum_{i>=1} A001358(i)*x^i.

cp's OEIS Frontend

A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

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A030010 Inverse Euler transform of primes.

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A072508 Decimal expansion of Backhouse constant.

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

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A247818 Decimal expansion of 1/(theta*P'(theta)), a constant appearing in the asymptotic evaluation of the coefficients q_n in 1/(1+P(x)), where P(x) is the generating function of the primes and theta the unique zero of P(x) in [-3/4, 0].

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A114041 Decimal expansion of -x, the real root of the power series with semiprime coefficients.

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Formula