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A253784 Numbers which have no two successive prime factors (when sorted into monotonic order) where the latter prime factor would be greater than the square of the former.

%I A253784 #15 Feb 19 2025 10:26:17
%S A253784 1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,21,23,24,25,27,29,30,31,32,
%T A253784 35,36,37,41,42,43,45,47,48,49,53,54,55,59,60,61,63,64,65,67,71,72,73,
%U A253784 75,77,79,81,83,84,85,89,90,91,95,96,97,101,103,105,107,108,109,113,115,119,120,121,125,126,127,128,131,133,135,137,139,143,144
%N A253784 Numbers which have no two successive prime factors (when sorted into monotonic order) where the latter prime factor would be greater than the square of the former.
%C A253784 In other words, {1} together with primes and such composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is less than the square of the previous: (p_i)^2 > p_j, (p_j)^2 > p_k, etc.
%C A253784 Whenever gcd(a(i),a(j)) > 1, then a(i)*a(j) and lcm(a(i),a(j)) are also members of this sequence.
%H A253784 Antti Karttunen, <a href="/A253784/b253784.txt">Table of n, a(n) for n = 1..10000</a>
%e A253784 1 is present as it has an empty prime factorization.
%e A253784 2 like all primes is present.
%e A253784 4 = 2*2 is present as 2^2 > 2.
%e A253784 9 = 3*3 is present as 3^2 > 3.
%e A253784 10 = 2*5 is NOT present, as 2^2 < 5.
%e A253784 30 = 2*3*5 is present, as 2^2 > 3 and 3^2 > 5.
%o A253784 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A253784 (define A253784 (MATCHING-POS 1 1 (lambda (n) (numbers-densely-distributed? (ifactor n)))))
%o A253784 (define (numbers-densely-distributed? lista) (cond ((null? lista) #t) ((null? (cdr lista)) #t) ((< (A000290 (car lista)) (cadr lista)) #f) (else (numbers-densely-distributed? (cdr lista)))))
%Y A253784 Complement: A253785.
%Y A253784 Subsequences: A251726 (a(n+1) differs from A251726(n) for the first time at n=23, where a(24) = 30, while A251726(23) = 31), A251728 (semiprimes only).
%Y A253784 Subsequence of A253567.
%Y A253784 Cf. A000290.
%K A253784 nonn
%O A253784 1,2
%A A253784 _Antti Karttunen_, Jan 16 2015