A327877 Numbers having an odd number of non-unitary prime factors.

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168

Offset: 1

Author

Amiram Eldar, Sep 28 2019

Keywords

• nonn

Comments

Differs from A190641(n) from n = 310 (900, the least number with 3 non-unitary prime factors, is in this sequence but not in A190641).

The asymptotic density of the numbers in this sequence is 0.338682... = 1 - A065493.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2, (1964), pp. 214-227.
• Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107, No. 1 (1933), pp. 188-232.
• I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330.

Crossrefs

Cf. A056170, A065493, A190641, A333634 (complement).

Programs

Mathematica

A056170[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Select[Range[200], OddQ[A056170[#]] &] (* after Jean-François Alcover at A056170 *)

PARI

\\ here b(n) is A056170(n)
b(n)={my(f=factor(n)[, 2]); sum(i=1, #f, f[i]>1)}
{ select(k->b(k)%2, [1..200]) } \\ Andrew Howroyd, Sep 28 2019