A338542 Numbers having exactly five non-unitary prime factors.

5336100, 7452900, 10672200, 12744900, 14905800, 15920100, 16008300, 18404100, 21344400, 22358700, 23328900, 25489800, 26680500, 29811600, 31472100, 31840200, 32016600, 36072036, 36808200, 37088100, 37264500, 37352700, 38234700, 39312900, 42380100, 42688800, 43956900

Offset: 1

Author

Amiram Eldar, Nov 01 2020

Keywords

• nonn

Comments

Numbers k such that A056170(k) = A001221(A057521(k)) = 5.

Numbers divisible by the squares of exactly five distinct primes.

The asymptotic density of this sequence is (eta_1^5 - 10*eta_1^3*eta_2 + 15*eta_1*eta_2^2 + 20*eta_1^2*eta_3 - 20*eta_2*eta_3 - 30*eta_1*eta_4 + 24*eta_5)/(20*Pi^2) = 0.0000015673..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

Examples

			5336100 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 is a term since it has exactly 5 prime factors, 2, 3, 5, 7 and 11, that are non-unitary.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Crossrefs

Subsequence of A013929, A318720 and A327877.

Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338541.

Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

Programs

Mathematica

Select[Range[2*10^7], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 5 &]