A381736 Integers k = p*q*r, where p < q < r are distinct primes and p*q > r.

30, 70, 105, 154, 165, 182, 195, 231, 273, 286, 357, 374, 385, 399, 418, 429, 442, 455, 494, 561, 595, 598, 627, 646, 663, 665, 715, 741, 759, 782, 805, 874, 897, 935, 957, 969, 986, 1001, 1015, 1023, 1045, 1054, 1085, 1102, 1105, 1131, 1173, 1178, 1209

Offset: 1

Author

Matthew Goers, Mar 05 2025

Keywords

• nonn

Comments

These are squarefree 3-almost-primes, called sphenic numbers, that are greater than the square of the largest of its prime factors. As all sphenic numbers are, by definition, less than the cube of their largest prime factor, numbers in this sequence satisfy r^2 < k < r^3, where k = p*q*r, p < q < r.

Examples

			30 = 2*3*5 and 2*3 > 5, so 30 is in the sequence.
70 = 2*5*7 and 2*5 > 7, so 70 is in the sequence.
110 = 2*5*11 but 2*5 < 11, so 110 is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Matthew Goers, Factors of Terms

Crossrefs

Intersection of A007304 (sphenic numbers) and A164596.

Cf. A382022.

Programs

Maple

N:= 2000: # for terms < N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
R:= NULL:
for k from 1 to nops(P) do
  for i from 1 to k-2 while P[i]*P[i+1]*P[k] < N do
     jmin:= max(i+1,ListTools:-BinaryPlace(P,P[k]/P[i])+1);
     jmax:= min(k-1,ListTools:-BinaryPlace(P,N/(P[i]*P[k])));
     R:= R, seq(P[i]*P[j]*P[k],j=jmin .. jmax);
od od:
sort([R]); # Robert Israel, Mar 28 2025

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]*f[[2, 1]] > f[[3, 1]]]; Select[Range[1500], q] (* Amiram Eldar, Mar 20 2025 *)

PARI

is_a381736(n) = my(F=factor(n)); omega(F)==3 && bigomega(F)==3 && F[1,1]*F[2,1]>F[3,1] \\ Hugo Pfoertner, Mar 08 2025

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A381736(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(min(x//(p*q),p*q-1))-b for a,p in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,q in enumerate(primerange(p+1,isqrt(x//p)+1),a+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025