A084256 -id:A084256 - cp's OEIS frontend

A247734 Decimal expansion of the coefficient c appearing in the asymptotic evaluation of the number of prime additive compositions of n as c*(1/xi)^n, where xi is A084256.

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

Offset: 0

Jean-François Alcover, Sep 23 2014

			0.3036552633952545488542057678902065632735...
1/xi = 1.4762287836208969657929439948482332947971...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Cf. A084256 (xi).

nMax = 200; digits = 102; f[x_] := Sum[x^Prime[n], {n, 1, nMax}]; fp[x_] := Sum[Prime[n]*x^(Prime[n] - 1), {n, 1, nMax}]; xi = x /. FindRoot[f[x] == 1, {x, 2/3}, WorkingPrecision -> digits+5]; c = 1/(xi*fp[xi]); RealDigits[c, 10, digits] // First

c = 1/(xi*f'(xi)), where f(x) is the sum over primes x^2 + x^3 + x^5 + x^7 + ..., xi (A084256) being the positive solution of f(x) = 1.

A084255 Decimal expansion of continued fraction 1/(2+1/(3+1/(5+1/(7+1/(11+...))))).

Original entry on oeis.org

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

Offset: 0

Frank Ellermann, May 23 2003

Decimal expansion of the constant whose continued fraction form is the sequence of all the prime numbers.

			0.4323320871859028689...

Michael Hartley, 0.4323320871859028689.., Prime Curios!
G. L. Honaker, Jr., Prime Curios! 0.4323320871859028689...
Marek Wolf, Continued fractions constructed from prime numbers, arxiv:1003.4015 [math.NT], 2010.

Cf. A000040, A084256, A084257, A152062

RealDigits[1/FromContinuedFraction@ Prime@ Range@ 34, 10, 111] (* or *)
RealDigits[ Fold[1/(#1 + #2) &, 1, Reverse@ Prime@ Range@35], 10, 111] (* Robert G. Wilson v, Dec 26 2016 *)

1/A064442. - Franklin T. Adams-Watters, Jul 31 2009

A084257 Decimal expansion of -x where x^2/2! + x^3/3! + x^5/5! + x^7/7! + x^11/11! + x^13/13! + ... = 0.

Original entry on oeis.org

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

Offset: 1

Frank Ellermann, May 23 2003

This is the only nonzero real solution.

			-2.307505293750382035...

Submitted to Prime Curios! by Michael Hartley.
G. L. Honaker, Jr., Prime Curios!: -2.307505293750382035...

Cf. A000040, A084256, A084256.

A078465 Primonacci numbers: a(n)=a(n-2)+a(n-3)+a(n-5)+a(n-7)+a(n-11)+...+a(n-p(k))+... until n <= p(k), where p(k) is the k-th prime. a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 12, 16, 26, 36, 55, 81, 118, 177, 257, 384, 564, 833, 1233, 1813, 2685, 3956, 5845, 8629, 12731, 18807, 27746, 40976, 60481, 89282, 131816, 194562, 287253, 424018, 625968, 924077

Offset: 1

Miklos Kristof, Jan 02 2003

a(n)/a(n-1) -> 1.476229...=1/x, where x satisfies the Sum x^p(n)=1 equation, i.e. x^2+x^3+x^5+x^7+x^11+... =1. (What constant is it?)

			a(12) = 36 = a(12-2)+a(12-3)+a(12-5)+a(12-7)+a(12-11) = a(10)+a(9)+a(7)+a(5)+a(1) = 16+12+5+2+1 = 36.

T. D. Noe, Table of n, a(n) for n=1..500

Cf. A078974 (the constant 1.47622...), A084256 (the constant 1/1.47622...)

import Data.List (genericIndex)
a078465 n = a078465_list `genericIndex` (n-1)
a078465_list = 1 : 1 : f 3 where
   f x = (sum $ map (a078465 . (x -)) $
         takeWhile (< x) a000040_list) : f (x + 1)
-- Reinhard Zumkeller, Jul 20 2012

a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[n - Prime[k]], {k, 1, PrimePi[n]}]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Mar 22 2011 *)

Name corrected by Sean A. Irvine, Jul 01 2025

A078974 Decimal expansion of constant C such that Sum_{k>=1} 1/C^p(k) = 1 where p(k) is the k-th prime.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 12 2003

			1.47622878362089696579294399484823329479712276085059327075519...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's Composition Constant, p. 293.

Cf. A078465, A084256.

RealDigits[x/.FindRoot[Sum[1/x^Prime[k], {k,1,120}] == 1, {x, 1.476}, WorkingPrecision -> 120]][[1, 1 ;; 105]] (* Jean-François Alcover, Mar 22 2011 *)

Equals 1/A084256.

A272037 Decimal expansion of x such that x + x^4 + x^9 + x^16 + x^25 + x^36 + ... = 1.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 18 2016

This constant is an analog of A084256 where primes are replaced with squares.

			0.705346681379806989636379706393941505260078161512292870517426781...

Eric Weisstein's MathWorld, Jacobi Theta Functions

Cf. A000122, A084256.

FindRoot[Sum[x^n^2, {n, 1, 100}] == 1, {x, 7/10}, WorkingPrecision -> 100][[1, 2]] // RealDigits // First
(* or *)
FindRoot[EllipticTheta[3, 0, x] == 3, {x, 7/10}, WorkingPrecision -> 100][[1, 2]] // RealDigits // First

solve(x=.7,.8,suminf(y=1,x^y^2)-1) \\ Charles R Greathouse IV, Apr 25 2016

Solution to theta_3(0,x) = 3, where theta_3 is the 3rd elliptic theta function.

a(99) corrected by Sean A. Irvine, Jul 24 2025

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A247734 Decimal expansion of the coefficient c appearing in the asymptotic evaluation of the number of prime additive compositions of n as c*(1/xi)^n, where xi is A084256.

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A084255 Decimal expansion of continued fraction 1/(2+1/(3+1/(5+1/(7+1/(11+...))))).

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A084257 Decimal expansion of -x where x^2/2! + x^3/3! + x^5/5! + x^7/7! + x^11/11! + x^13/13! + ... = 0.

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A078465 Primonacci numbers: a(n)=a(n-2)+a(n-3)+a(n-5)+a(n-7)+a(n-11)+...+a(n-p(k))+... until n <= p(k), where p(k) is the k-th prime. a(1)=a(2)=1.

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A078974 Decimal expansion of constant C such that Sum_{k>=1} 1/C^p(k) = 1 where p(k) is the k-th prime.

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A272037 Decimal expansion of x such that x + x^4 + x^9 + x^16 + x^25 + x^36 + ... = 1.

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