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A349241 Numbers N = pqrs such that |pqr - s| > |ps - qr|, where p <= q <= r <= s are the 4 prime factors of N.

%I A349241 #19 Nov 14 2021 03:30:43
%S A349241 16,24,36,54,60,81,90,100,126,135,140,150,189,196,210,225,250,294,308,
%T A349241 315,330,350,364,375,390,441,462,484,490,495,525,546,550,572,585,625,
%U A349241 650,676,686,693,714,726,735,748,770,798,819,825,836,850,858,875,884,910,950,975,988
%N A349241 Numbers N = pqrs such that |pqr - s| > |ps - qr|, where p <= q <= r <= s are the 4 prime factors of N.
%C A349241 The set A014613 of numbers n with bigomega(n) = A001222(n) = 4, can be partitioned in these here and their complement A349242. It was suggested (cf. math-fun post in LINKS) to call these here the "trans"- and the others the "cis"-type.
%C A349241 These here include squares of semiprimes (A074985), and in particular 4th powers of primes (A030514), for which |ps - qr| = 0.
%C A349241 Within the 4-almost primes below 10^k, k = 2, 3, ...,8, we have (8, 57, 497, 4960, 49228, 491397, 4869917, ...) of trans type, and more than twice (or even three times) as many of cis type.
%H A349241 Marc LeBrun, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/message/PTOR2R5FAKLHLW6M2HKKXNXDHTQ4VIUR/">four factor fun</a>, math-fun mailing list (available for subscribers only), Nov 10 2021
%F A349241 { N in A014613 | |g - N/g| > |sg - N/sg| }, where g = gpf(N) = A006530(N) is the greatest, and s = spf(N) = A020639(N) is the smallest prime factor.
%e A349241 16 = 2^4 = u*v with u = v = 2*2 closer (equal) than u = 2*2*2, v = 2 (difference 8 - 2 = 6).
%e A349241 24 = 2^3*3 = u*v with u = 2*2, v = 2*3 closer (distance 6 - 4 = 2) than u = 2*2*2, v = 3 (distance 8 - 3 = 5).
%e A349241 36 = 2^2*3^2 = u*v with u = v = 2*3 closer (equal) than u = 2^2*3, v = 3 (difference 12 - 3 = 9).
%e A349241 The 4-almost prime 40 = 2^3*5 is not in this sequence because the factorization 40 = u*v with u = 2^3, v = 5 has closer factors (distance 8 - 5 = 3) than u = 2*2, v = 2*5 (distance 10 - 4 = 6).
%o A349241 (PARI) select( {is_A349241(n,a(u)=abs(u-n\u))=bigomega(n)==4 && a((s=factor(n)[,1])[#s])>a(s[1]*s[#s])}, [1..1000])
%o A349241 (Python)
%o A349241 from itertools import chain
%o A349241 from sympy import factorint
%o A349241 def expand(n):
%o A349241     return list(chain.from_iterable([[i[0] for j in range(i[1])] for i in factorint(n).items()]))
%o A349241 def is_ok(p,q,r,s):
%o A349241     return abs(p*q*r-s) > abs(p*s-q*r)
%o A349241 print([i for  i in range(2, 1000) if len(expand(i)) == 4 and is_ok(*expand(i))]) # _Gleb Ivanov_, Nov 12 2021
%Y A349241 Cf. A014613, A349242, A006530, A020639.
%K A349241 nonn
%O A349241 1,1
%A A349241 _M. F. Hasler_, Nov 12 2021