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

16, 24, 36, 54, 60, 81, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 308, 315, 330, 350, 364, 375, 390, 441, 462, 484, 490, 495, 525, 546, 550, 572, 585, 625, 650, 676, 686, 693, 714, 726, 735, 748, 770, 798, 819, 825, 836, 850, 858, 875, 884, 910, 950, 975, 988

Offset

1,1

Comments

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.

These here include squares of semiprimes (A074985), and in particular 4th powers of primes (A030514), for which |ps - qr| = 0.

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.

Links

Table of n, a(n) for n=1..57.

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

16 = 2^4 = u*v with u = v = 2*2 closer (equal) than u = 2*2*2, v = 2 (difference 8 - 2 = 6).
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).
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).
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).

Prog

(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])
(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, A349242, A006530, A020639.

Sequence in context: A103248 A286195 A140135 * A120142 A110228 A269519

Adjacent sequences: A349238 A349239 A349240 * A349242 A349243 A349244

Keyword

nonn

Author

M. F. Hasler, Nov 12 2021

Status

approved