A136617 a(n) = largest k such that the sum of k consecutive reciprocals 1/n + ... + 1/(n+k-1) does not exceed 1.

1, 2, 4, 6, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 24, 26, 28, 30, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 67, 69, 71, 73, 74, 76, 78, 79, 81, 83, 85, 86, 88, 90, 91, 93, 95, 97, 98, 100, 102, 103, 105, 107, 109, 110, 112, 114, 115

Offset

1,2

Comments

Heuristic formula from David Cantrell (SeqFan mailing list, January 2008). Think of a ruler with harmonic numbers H(n) as marks. Then A136617(n) gives the number of marks m-n+1 = A136616(n)-n+1:

.............H........H.....H........***.....H.......

..............n-1......n.....n+1..............m......

...........----o-------+------+-----.***.-----+-o----

................\______________..______________/......

...............................\/.....................

............................Length 1..................

The first 23 terms of A083088 are identical to those of A136617 but the limits of A083088(n)/n and A136617(n)/n for n->oo are different.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.

Formula

a(n) = A136616(n-1) - n + 1 with David Cantrell's heuristics: a(n) = floor( (e - 1)*(n - 1/2) + (e - 1/e)/(24*(n - 1/2)) ).

Example

a(3) = 4 because 1/3+1/4+1/5+1/6 < 1 has 4 summands; adding 1/7 exceeds 1.

Maple

A136617 := proc(n) local t, m; t:= 0; for m from n do t:= t+1/m; if t > 1 then return m-n; fi; od; end proc; [seq(A136617(n), n=1..100)]; # Robert Israel, January 2008

Mathematica

Table[Module[{start = Floor[z (E - 1)] - 1},
  NestWhile[# + 1 &, start, HarmonicNumber[# + z] - HarmonicNumber[z] + 1/z <= 1 &]], {z, 1, 100}] (* Peter J. C. Moses, Aug 20 2012 *)

Crossrefs

Cf. A136616, A002387, A004080, A079353, A081881, A096618, A103762, A118050, A118051.

Sequence in context: A080755 A372779 A083089 * A275814 A285376 A198081

Adjacent sequences: A136614 A136615 A136616 * A136618 A136619 A136620

Keyword

easy,nonn

Author

Rainer Rosenthal, Jan 13 2008

Status

approved