OFFSET
1,1
COMMENTS
Prime signature of a(n) is 2 followed by at least one 1.
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
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
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).
MATHEMATICA
Select[Select[Range[1, 1100, 2], PrimeOmega[#] > PrimeNu[#] > 1 &], And[OddQ[#1], #1/(Times @@ #2) == #2[[1]]] & @@ {#, FactorInteger[#][[All, 1]]} &]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Jan 05 2024
STATUS
approved