A349242 Numbers N = pqrs such that |pqr - s| < |ps - qr|, where p <= q <= r <= s are the 4 prime factors of N.

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

Table of n, a(n) for n=1..51.

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

Cf. A014613, A349241, A006530, A020639.

Sequence in context: A109730 A100333 A372298 * A369150 A260609 A260602

Adjacent sequences: A349239 A349240 A349241 * A349243 A349244 A349245

Keyword

nonn

Author

M. F. Hasler, Nov 12 2021

Status

approved