A061117 -id:A061117 - cp's OEIS frontend

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

Jeff Burch

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

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

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

Cf. A061117.

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

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 *)

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

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

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

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

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

Definition clarified by Jonathan Sondow, Jan 23 2014

A061112 a(n) is the minimum number of divisors for any composite between prime(n) and prime(n+1).

Original entry on oeis.org

3, 4, 3, 6, 4, 6, 4, 3, 8, 4, 4, 8, 4, 3, 4, 12, 4, 4, 12, 4, 4, 4, 4, 6, 8, 4, 12, 4, 3, 4, 4, 8, 4, 12, 4, 4, 4, 3, 4, 18, 4, 14, 4, 12, 4, 4, 4, 12, 8, 4, 20, 4, 4, 4, 4, 16, 4, 4, 8, 3, 4, 4, 16, 4, 4, 4, 4, 12, 8, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 24, 4, 20, 4, 8, 4, 4, 4, 16, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 2

Labos Elemer, May 29 2001

nonn

			prime(30)=113 is followed by 13 composites; their numbers of divisors are {8, 4, 6, 6, 4, 4, 16, 3, 4, 4, 6, 4, 12}. The smallest is 3, so a(30)=3. [corrected by _Jon E. Schoenfield_, Sep 20 2022]

Harry J. Smith, Table of n, a(n) for n = 2..1000.

Cf. A000005, A061117.

Array[Min@ DivisorSigma[0, Range[#1 + 1, #2 - 1]] & @@ Prime[# + {0, 1}] &, 95, 2] (* Michael De Vlieger, Aug 10 2023 *)

{ n=-1; q=3; m=10^6; forprime (p=5, prime(1003), a=m; for (i=q + 1, p - 1, a=min(numdiv(i), a)); q=p; write("b061112.txt", n++, " ", a) ) } \\ Harry J. Smith, Jul 19 2009

a(n) = Min_{c=prime(n)+1..prime(n+1)-1} tau(c) where tau(c) is the number of divisors of c.

cp's OEIS Frontend

A030515 Numbers with exactly 6 divisors.

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