login
Odd terms in A366825.
3

%I #27 Jan 08 2024 01:39:21

%S 45,63,99,117,153,171,175,207,261,275,279,315,325,333,369,387,423,425,

%T 475,477,495,531,539,549,575,585,603,637,639,657,693,711,725,747,765,

%U 775,801,819,833,855,873,909,925,927,931,963,981,1017,1025,1035,1071,1075

%N Odd terms in A366825.

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

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

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

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

%C 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

%H Michael De Vlieger, <a href="/A366460/b366460.txt">Table of n, a(n) for n = 1..10000</a>

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

%e a(1) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.

%e a(2) = 63 = 9*7 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(7), i.e., 3 < 7.

%e Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

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

%o (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

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

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jan 05 2024