A366460 Odd terms in A366825.

45, 63, 99, 117, 153, 171, 175, 207, 261, 275, 279, 315, 325, 333, 369, 387, 423, 425, 475, 477, 495, 531, 539, 549, 575, 585, 603, 637, 639, 657, 693, 711, 725, 747, 765, 775, 801, 819, 833, 855, 873, 909, 925, 927, 931, 963, 981, 1017, 1025, 1035, 1071, 1075

Offset: 1

Author

Michael De Vlieger, Jan 05 2024

Keywords

• nonn
• easy

Comments

Proper subset of A364997, in turn a proper subset of A364996, which is a proper subset of A126706.

Prime signature of a(n) is 2 followed by at least one 1.

Numbers of the form A065642(k) where k is an odd term in A120944.

Numbers of the form p^2 * m, squarefree m > 1, odd prime p < lpf(m), where lpf(m) = A020639(m).

The asymptotic density of this sequence is (2/(3*Pi^2)) * Sum_{p odd prime} ((1/p^2) * (Product_{odd primes q <= p} (q/(q+1)))) = 0.0537475047... . - Amiram Eldar, Jan 08 2024

Examples

			a(1) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
a(2) = 63 = 9*7 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(7), i.e., 3 < 7.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007947, A020639, A065642, A120944, A126706, A364996, A364997, A364999.

Programs

Mathematica

Select[Select[Range[1, 1100, 2], PrimeOmega[#] > PrimeNu[#] > 1 &], And[OddQ[#1], #1/(Times @@ #2) == #2[[1]]] & @@ {#, FactorInteger[#][[All, 1]]} &]

PARI

is(n) = {my(e); n%2 && e = factor(n)[, 2]; #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1; } \\ Amiram Eldar, Jan 08 2024

Formula

{a(n)} = {A366825 \ A364999}.