A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

8, 62, 454, 3348, 24734, 182760, 1350428, 9978382, 73730824, 544801200, 4025566630, 29745137662, 219788490858, 1624029488844, 12000044999386, 88669005690160, 655180257281000, 4841163675961122, 35771629985782052

Offset: 1

Author

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

Keywords

• nonn

Comments

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:

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

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

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

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

References

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

Crossrefs

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

Cf. A056054.

Programs

Mathematica

s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 12}]
(* or assuming that the Mathematica coding in A002387 is correct then *)
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 *)

Formula

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

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