A072508 -id:A072508 - cp's OEIS frontend

A088751 Decimal expansion of -x, the real root of the equation 0 = 1 + Sum_{k>=1} prime(k) x^k. The inverse of Backhouse's constant (A072508).

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

Offset: 0

T. D. Noe, Oct 14 2003

This constant is computed in Finch's article. This number is easier to compute than Backhouse's constant. Except for an additional term of 0, the continued fraction expansion is the same as that of Backhouse's constant.

			0.68677783446063...

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]
Eric Weisstein's World of Mathematics, Backhouse's Constant.

Cf. A030010, A072508.

RealDigits[ -x/.FindRoot[0==1+Sum[x^n Prime[n], {n, 1000}], {x, {0, 1}}, WorkingPrecision->100]][[1]]

A074269 Continued fraction for the Backhouse constant (A072508).

Original entry on oeis.org

1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, 3, 12, 1, 2, 2, 1, 1, 15, 1, 1, 1, 2, 2, 1, 4, 5, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 6, 1, 22, 3, 11, 2, 1, 13, 2, 7, 2, 5, 2, 17, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 14, 551, 1, 1, 1, 1, 8, 2, 28, 1, 5, 6

Offset: 0

Robert G. Wilson v, Sep 20 2002

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]
Eric Weisstein's World of Mathematics, Backhouse's Constant

Cf. A072508 (decimal expansion).

(* down load the constant from the link above & set it equal to a *) ContinuedFraction[a, 100]

Offset changed by Andrew Howroyd, Jul 06 2024

A030018 Coefficients in 1/(1+P(x)), where P(x) is the generating function of the primes.

Original entry on oeis.org

1, -2, 1, -1, 2, -3, 7, -10, 13, -21, 26, -33, 53, -80, 127, -193, 254, -355, 527, -764, 1149, -1699, 2436, -3563, 5133, -7352, 10819, -15863, 23162, -33887, 48969, -70936, 103571, -150715, 219844, -320973, 466641, -679232, 988627, -1437185, 2094446, -3052743

Offset: 0

N. J. A. Sloane

sign

a(n+1)/a(n) => ~-1.456074948582689671... (see A072508). - Zak Seidov, Oct 01 2011

Seiichi Manyama, Table of n, a(n) for n = 0..6124 (terms 0..1000 from Zak Seidov)
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Backhouse's Constant

Cf. A000040, A072508, A340991.

a:= proc(n) option remember; `if`(n=0, 1,
      -add(ithprime(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 13 2018

max = 50; P[x_] := 1 + Sum[Prime[n]*x^n, {n, 1, max}]; s = Series[1/P[x], {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Sep 24 2014 *)

v=[];for(n=1,50,v=concat(v,-prime(n)-sum(i=1,n-1,prime(i)*v[#v-i+1])));v \\ Derek Orr, Apr 28 2015

Apply inverse of "INVERT" transform to primes: INVERT: a's from b's in 1+Sum a_i x^i = 1/(1-Sum b_i x^i).
a(n) = -prime(n) - Sum_{i=1..n-1} prime(i)*a(n-i), for n > 0. - Derek Orr, Apr 28 2015
a(n) = Sum_{k=0..n} (-1)^k * A340991(n,k). - Alois P. Heinz, Feb 01 2021

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

sign

			(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)

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.

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

Original entry on oeis.org

-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

cp's OEIS Frontend

A088751 Decimal expansion of -x, the real root of the equation 0 = 1 + Sum_{k>=1} prime(k) x^k. The inverse of Backhouse's constant (A072508).

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A074269 Continued fraction for the Backhouse constant (A072508).

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A030018 Coefficients in 1/(1+P(x)), where P(x) is the generating function of the primes.

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A030010 Inverse Euler transform of primes.

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

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