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.

906180359, 906180361, 906180363, 906180365, 906180367, 906180369, 906180371, 906180373, 906180375, 906180391, 906180393, 906180423, 906180425, 906180517, 906180519, 906180525, 906180529, 906180533, 906180537, 906180553, 906180555

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