OFFSET
1,1
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Matthew Goers, Factors of Terms
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Goers, Mar 05 2025
STATUS
approved