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%I #4 Feb 11 2014 19:05:31
%S 906180359,906180361,906180363,906180365,906180367,906180369,
%T 906180371,906180373,906180375,906180391,906180393,906180423,
%U 906180425,906180517,906180519,906180525,906180529,906180533,906180537,906180553,906180555
%N Numbers n such that O(n) = E(n)-1, where O(n), E(n) denote the number of integers k, 1 <= k <= n, such that Omega(n) is even or odd, respectively and Omega(n) denotes the number of prime factors of n, counting multiplicity.
%C In 1919, George Polya conjectured that O(n) >= E(n) for n >= 2. However, in 1958, C. B. Haselgrove showed that there are infinitely many positive integers n for which O(n) < E(n). In 1966, R.S. Lehman showed that 906180359 is the smallest positive integer for which O(n) = E(n) - 1. (Tattersall, p. 92)
%D J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001.
%t (*This program relies on Lehman's result that 906180359 is the least term of the sequence.*) Omega[n_] := Apply[Plus, Transpose[FactorInteger[n]][[2]]]; start = 906180360; l = {906180359}; o = 0; e = 1; i = start; While[i < 906193475, If[Mod[Omega[i], 2] == 0, e = e + 1, o = o + 1]; If[o == e - 1, l = Append[l, i]]; i = i + 1]; l
%K nonn
%O 1,1
%A _Joseph L. Pe_, Oct 30 2002