A297401 Non-sphenic numbers with exactly 8 divisors.

24, 40, 54, 56, 88, 104, 128, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 459, 472, 488, 513, 536, 568, 584, 621, 632, 664, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029, 1048, 1096, 1107, 1112, 1161, 1192

Offset: 1

Author

G. L. Honaker, Jr., Dec 29 2017

Keywords

• nonn

Comments

These are the numbers of the form p^3*q (with primes p and q distinct) or p^7. Thus it is the union of A065036 and A092759, and this can be used for direct enumeration. - Alex Meiburg, Dec 31 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios!
• Wikipedia, Sphenic number

Crossrefs

Subsequence of A030626.

Cf. A000005, A030626, A065036, A092759.

Programs

Maple

N:= 1000: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..N)]):
R:= NULL:
for p in P do
  if p^7 <= N then R:= R, p^7 fi;
  if p^3 > N then break fi;
  for q in P while p^3*q <= N do if q <> p then R:= R, p^3*q fi od:
od:
sort([R]); # Robert Israel, Dec 31 2017

Mathematica

Select[Range@ 1200, And[DivisorSigma[0, #] == 8, Nand[PrimeNu[#] == 3, PrimeOmega[#] == 3]] &] (* Michael De Vlieger, Dec 29 2017 *)

PARI

isok(n) = !((bigomega(n)==3) && (omega(n)==3)) && (numdiv(n) == 8); \\ Michel Marcus, Dec 29 2017

Python

from sympy import primepi, primerange, integer_nthroot
def A297401(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 int(n+x-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])-primepi(integer_nthroot(x,7)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Equals {A030626} \ {A007304}. - Omar E. Pol, Dec 30 2017

Extensions

More terms from Michel Marcus, Dec 29 2017