A056054 - OEIS

%I #11 Jan 05 2025 23:42:18

%S 8,62,454,3348,24734,182760,1350428,9978382,73730824,544801200,

%T 4025566630,29745137662,219788490858,1624029488844,12000044999386,

%U 88669005690160,655180257281000,4841163675961122,35771629985782052

%N a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

%C Numbers 2*m such that floor(f(m)) = floor(f(m-1)) where f(m) = Sum_{j=1..m} ((2*j-1)/(2*j)). Examples:

%C   floor(f(1))=floor(1/2)=0;

%C   floor(f(2))=floor(1/2+3/4)=floor(1.25)=1, then 2*2=4 is not in the sequence;

%C   floor(f(3))=floor(1/2+3/4+5/6)=floor(2.083..)=2, then 2*3=6 is not in the sequence;

%C   floor(f(4))=floor(1/2+3/4+5/6+7/8)=floor(2.958..)=2, then 2*4=8 is the first term of the sequence. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 15 2007

%D Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64.

%F a(n) = 2*A002387(2n).

%F The next term is approximately the previous term * e^2.

%t s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 12}]

%t (* or assuming that the Mathematica coding in A002387 is correct then *)

%t b[n_] := Module[{k = Floor[2a[2n]]}, If[ EvenQ[k], k, k + 1]]; Table[ b[n], {n, 19}] (* _Robert G. Wilson v_, Apr 17 2004 *)

%Y Cf. A002387, A056053, A091463, A091464, A091465.

%Y Cf. A056054.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Jul 25 2000 and Jan 11 2004