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A136617 a(n) = largest k such that the sum of k consecutive reciprocals 1/n + ... + 1/(n+k-1) does not exceed 1. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A083088 A080755 A083089 * A275814 A285376 A198081

Adjacent sequences:  A136614 A136615 A136616 * A136618 A136619 A136620

KEYWORD

easy,nonn

AUTHOR

Rainer Rosenthal, Jan 13 2008

STATUS

approved

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Last modified October 16 09:02 EDT 2021. Contains 348041 sequences. (Running on oeis4.)