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A349242
Numbers N = pqrs such that |pqr - s| < |ps - qr|, where p <= q <= r <= s are the 4 prime factors of N.
1
40, 56, 84, 88, 104, 132, 136, 152, 156, 184, 198, 204, 220, 228, 232, 234, 248, 260, 276, 296, 297, 306, 328, 340, 342, 344, 348, 351, 372, 376, 380, 414, 424, 444, 459, 460, 472, 476, 488, 492, 510, 513, 516, 522, 532, 536, 558, 564, 568, 570, 580
OFFSET
1,1
COMMENTS
The set A014613 of numbers n with bigomega(n) = A001222(n) = 4, can be partitioned in these here (of "cis" type), and their complement A349241, those of "trans" type; see the math-fun post in LINKS for this suggested terminology.
Within the 4-almost primes below 10^k, k = 2, 3, ...,8, we have (92, 1215, 13784, 148834, 1559299, 16089405, ...) of cis type, more than twice as much as those of trans type.
LINKS
Marc LeBrun, four factor fun, math-fun mailing list (available for subscribers only), Nov 10 2021
FORMULA
{ 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.
EXAMPLE
The first three 4-almost primes 16 = 2^4, 24 = 2^3*3 and 36 = 2^2*3^2 are not in this sequence, because the factorization u*v with closest possible factors u and v consists of choosing both, u and v, to be semiprimes (namely, 4*4, 4*6 and 6*6).
The 4-almost prime 40 = 2^3*5 is 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).
PROG
(PARI) select( {is_A349242(n, a(u)=abs(u-n\u))=bigomega(n)==4 && a((s=factor(n)[, 1])[#s])<a(s[1]*s[#s])}, [1..1000])
(Python)
from itertools import chain
from sympy import factorint
def expand(n):
return list(chain.from_iterable([[i[0] for j in range(i[1])] for i in factorint(n).items()]))
def is_ok(p, q, r, s):
return abs(p*q*r - s)<abs(p*s-q*r)
print([i for i in range(2, 1000) if len(expand(i)) == 4 and is_ok(*expand(i))]) #Gleb Ivanov, Nov 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 12 2021
STATUS
approved