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A076135
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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.
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0
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906180359, 906180361, 906180363, 906180365, 906180367, 906180369, 906180371, 906180373, 906180375, 906180391, 906180393, 906180423, 906180425, 906180517, 906180519, 906180525, 906180529, 906180533, 906180537, 906180553, 906180555
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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)
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REFERENCES
| J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001.
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MATHEMATICA
| (*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
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CROSSREFS
| Sequence in context: A157798 A189229 A051470 * A015382 A115385 A186805
Adjacent sequences: A076132 A076133 A076134 * A076136 A076137 A076138
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KEYWORD
| nonn
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AUTHOR
| Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Oct 30 2002
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