A338540 -id:A338540 - cp's OEIS frontend

A338539 Numbers having exactly two non-unitary prime factors.

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 2.
Numbers divisible by the squares of exactly two distinct primes.
Subsequence of A036785 and first differs from it at n = 44.
The asymptotic density of this sequence is (3/Pi^2)*(eta_1^2 - eta_2) = 0.0532928864..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			36 = 2^2 * 3^2 is a term since it has exactly 2 prime factors, 2 and 3, that are non-unitary.

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.

Subsequence of A013929 and A036785.
Cf. A001221, A056170, A057521, A190641, A338540, A338541, A338542, A347892 (subsequence).
Cf. A154945 (eta_1), A324833 (eta_2).

Select[Range[1000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 2 &]

A338541 Numbers having exactly four non-unitary prime factors.

Original entry on oeis.org

44100, 88200, 108900, 132300, 152100, 176400, 213444, 217800, 220500, 260100, 264600, 298116, 304200, 308700, 324900, 326700, 352800, 396900, 426888, 435600, 441000, 456300, 476100, 485100, 509796, 520200, 529200, 544500, 573300, 592900, 596232, 608400, 617400

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 4.
Numbers divisible by the squares of exactly four distinct primes.
The asymptotic density of this sequence is (eta_1^4 - 6*eta_1^2*eta_2 + 3*eta_2^2 + 8*eta_1*eta_3 - 6*eta_4)/(4*Pi^2) = 0.0000970457..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			44100 = 2^2 * 3^2 * 5^2 * 7^2 is a term since it has exactly 4 prime factors, 2, 3, 5 and 7, that are non-unitary.

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.

Subsequence of A013929 and A318720.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4).

Select[Range[620000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 4 &]

A338542 Numbers having exactly five non-unitary prime factors.

Original entry on oeis.org

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

Amiram Eldar, Nov 01 2020

nonn

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).

			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.

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.

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).

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

cp's OEIS Frontend

A338539 Numbers having exactly two non-unitary prime factors.

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A338541 Numbers having exactly four non-unitary prime factors.

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A338542 Numbers having exactly five non-unitary prime factors.

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