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A076135 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. 0
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; text; internal format)
OFFSET

1,1

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)

REFERENCES

J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001.

LINKS

Table of n, a(n) for n=1..21.

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

CROSSREFS

Sequence in context: A328135 A189229 A051470 * A015382 A115385 A186805

Adjacent sequences:  A076132 A076133 A076134 * A076136 A076137 A076138

KEYWORD

nonn

AUTHOR

Joseph L. Pe, Oct 30 2002

STATUS

approved

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Last modified May 14 03:33 EDT 2021. Contains 343872 sequences. (Running on oeis4.)