A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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

Offset: 1

Author

Jaroslav Krizek, Jul 16 2013

Keywords

• nonn
• cons

Comments

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015

Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...

Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.

Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).

Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).

Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.

e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.

1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

Examples

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Engel Expansion
• Googology Wiki, Double factorial

Crossrefs

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

Programs

Magma

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

Mathematica

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

PARI

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

Sage

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019