A030515 Numbers with exactly 6 divisors.

12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

Numbers which are either the 5th power of a prime or the product of a prime and the square of a different prime, i.e., numbers which are in A050997 (5th powers of primes) or A054753. - Henry Bottomley, Apr 25 2000

Also numbers which are the square root of the product of their proper divisors. - Amarnath Murthy, Apr 21 2001

Such numbers are multiplicatively 3-perfect (i.e., the product of divisors of a(n) equals a(n)^3). - Lekraj Beedassy, Jul 13 2005

Since A119479(6)=5, there are never more than 5 consecutive terms. Quintuples of consecutive terms start at 10093613546512321, 14414905793929921, 266667848769941521, ... (A141621). No such quintuple contains a term of the form p^5. - Ivan Neretin, Feb 08 2016

References

• Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.
• Eric Weisstein's World of Mathematics, Divisor Product

Crossrefs

Cf. A061117.

Programs

Maple

N:= 1000: # to get all terms <= N
Primes:= select(isprime, {2,seq(i,i=3..floor(N/4))}):
S:= select(`<=`,{seq(p^5, p = Primes),seq(seq(p*q^2, p=Primes minus {q}),q=Primes)},N):
sort(convert(S,list)); # Robert Israel, Feb 10 2016

Mathematica

f[n_]:=Length[Divisors[n]]==6; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
Select[Range[500],DivisorSigma[0,#]==6&] (* Harvey P. Dale, Oct 02 2014 *)

PARI

is(n)=numdiv(n)==6 \\ Charles R Greathouse IV, Jan 23 2014

Python

from sympy import divisor_count
def ok(n): return divisor_count(n) == 6
print([k for k in range(429) if ok(k)]) # Michael S. Branicky, Dec 18 2021

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030515(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(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Union of A050997 and A054753. - Lekraj Beedassy, Jul 13 2005

A000005(a(n))=6. - Juri-Stepan Gerasimov, Oct 10 2009

Extensions

Definition clarified by Jonathan Sondow, Jan 23 2014