OFFSET
1,1
COMMENTS
Since all the terms are even, the prime 2 appears with an exponent of 1 or 2: either 2 is the only prime appearing with an exponent of 2, or the unique prime appearing with an exponent of 2 is an odd prime.
The asymptotic density of this sequence is (2/Pi^2) * (Sum_{p prime} 1/(p*(p+1)) + 1/3) = A185197 * (A179119 + 1/3) = 0.1344660297679805099613... . - Amiram Eldar, Aug 10 2026
FORMULA
Numbers of the form p^2*q_1*q_2*...*q_r, where r >= 2, p,q_1,q_2,...,q_r are distinct primes, and at least one of these primes is 2.
Equivalently, the sequence is the disjoint union of numbers of the forms:
1. 4*s, where s is odd and squarefree and A001221(s) >= 2, and
2. 2*p^2*s, where p is an odd prime, s is odd and squarefree, gcd(p,s) = 1, and A001221(s) >= 1.
EXAMPLE
60 is a term because 60 = 2^2*3*5, so 2 is the unique prime occurring with exponent 2.
90 is a term because 90 = 2*3^2*5, so 3 is the unique prime occurring with exponent 2.
315 is in A085987 but not in this sequence, since 315 = 3^2*5*7 is odd.
420 is in this sequence but not in A085987, since 420 = 2^2*3*5*7 has four distinct prime factors.
MATHEMATICA
nn = 840; s = Select[Range[2, nn, 2], And[PrimeNu[#] > 2, SquareFreeQ[#]] &]; Union@ Flatten@ Map[Function[{k, w}, Map[If[# < nn, #, Nothing] &[k*#] &, w]] @@ {#, FactorInteger[#][[;; , 1]]} &, s] (* Michael De Vlieger, Aug 11 2026 *)
PROG
(PARI) isok(k) = if(k == 1 || k % 2, 0, my(e = factor(k)[ , 2]); if(#e < 3, 0, e = vecsort(e, , 4); e[1] == 2 && e[2] == 1)); \\ Amiram Eldar, Aug 10 2026
(PARI) isok(k) = my(f=factor(k), o=omega(f)); k%2 == 0 && o >= 3 && bigomega(f) == (o+1); \\ Michel Marcus, Aug 12 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Charles Kusniec, Aug 09 2026
STATUS
approved