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 A226161 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/2. 3
 1, 2, 3, 4, 7, 11, 19, 31, 51, 83, 137, 227, 373, 616, 1015, 1674, 2759, 4550, 7501, 12367, 20390, 33617, 55425, 91380, 150661, 248397, 409538, 675214, 1113239, 1835421, 3026097, 4989191, 8225785, 13562027, 22360003, 36865412, 60780790, 100210581, 165219316 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: a(n+1)/a(n) converges to 1.64872... The conjecture is correct, a(n+1)/a(n) ~ exp(1/2) (A019774). - Charles R Greathouse IV, Jun 03 2013 Conjecture: a(n) = round(exp(n/2-gamma)) for all n, where gamma is the Euler-Mascheroni constant (see A001620). - Jon E. Schoenfield, Jul 19 2015 The terms up to a(52) contained in the b-file have been obtained by working with quadruple-precision (128 bits) floating point numbers, hence there is a very small probability they are off by 1. - Giovanni Resta, Jul 21 2015 All terms in the b-file are correct. Moreover, the above conjecture that a(n) = round(exp(n/2-gamma)) has been verified for all n in 1..10000. - Jon E. Schoenfield, Jul 22 2015 LINKS Giovanni Resta, Table of n, a(n) for n = 1..52 EXAMPLE a(5) = 7 because 1 + 1/2 + ... + 1/6 < 5/2 < 1 + 1/2 + ... + 1/6 + 1/7. MATHEMATICA nn = 24; g = 1/2; f[n_] := 1/n; a[1] = 1; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) > n*g &], {n, nn}]; Map[a, Range[nn]] PROG (PARI) first(m)=my(v=vector(m), i, k); for(i=1, m, k=1; while(sum(x=1, k, 1/x)<=i/2, k++); v[i]=k; ); v; \\ Anders Hellström, Jul 19 2015 CROSSREFS Cf. A001620, A019774, A226160. Sequence in context: A080074 A317767 A018064 * A188624 A327010 A341823 Adjacent sequences: A226158 A226159 A226160 * A226162 A226163 A226164 KEYWORD nonn AUTHOR Clark Kimberling, May 29 2013 EXTENSIONS a(29)-a(35) from Jean-François Alcover, Jun 04 2013 a(36)-a(37) from Jon E. Schoenfield, Aug 31 2013 a(38)-a(39) from Jon E. Schoenfield, Jul 19 2015 STATUS approved

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Last modified December 7 01:51 EST 2022. Contains 358649 sequences. (Running on oeis4.)