%I #11 Jul 25 2024 20:58:15
%S 1,5,16,26,36,46,56,66,76,86,96,106,116,126,136,146,156,166,176,186,
%T 196,206,216,226,236,246,257,267,277,287,297,307,317,327,337,347,357,
%U 367,377,387,397,407,417,427,437,447,457,467,477,487,497,508,518,528
%N Numbers which occur exactly once in row 1 of the array at A227581.
%C It appears that every positive integer k occurs exactly once or exactly twice as a solution of H(n) - H(n+k) < g-1 < H(n) - H(n+k-1) as n runs through the positive integers, where H denotes harmonic number, and g denotes the Euler-Mascheroni constant. See A227581.
%H Clark Kimberling, <a href="/A227586/b227586.txt">Table of n, a(n) for n = 1..400</a>
%e That 5 occurs just once, and 4 and 6 each occur twice, corresponds to these inequalities:
%e H(6) - H(6 + 4) < g-1 < H(6) - H(6 + 3),
%e H(7) - H(7 + 4) < g-1 < H(7) - H(7 + 3),
%e H(8) - H(8 + 5) < g-1 < H(8) - H(8 + 4),
%e H(9) - H(9 + 6) < g-1 < H(9) - H(9 + 5),
%e H(10) - H(10 + 6) < g-1 < H(10) - H(10 + 5).
%t z = 1500; r[n_] := r[n] = Module[{Nn = N[n, 50]}, NestWhile[# + 1 &, Floor[(1 + n)/2], ! HarmonicNumber[1] + HarmonicNumber[Nn] - HarmonicNumber[Nn + #] < EulerGamma &]]; u[k_] := Length[Split[Table[r[n], {n, z}]][[k]]]; t = Table[u[k], {k, 1, z/2}]; Flatten[Position[t, 1]] (* A227586 *)
%Y Cf. A227581.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jul 17 2013