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A030515 Numbers with exactly 6 divisors.

%I A030515 #59 Feb 21 2025 13:07:37
%S A030515 12,18,20,28,32,44,45,50,52,63,68,75,76,92,98,99,116,117,124,147,148,
%T A030515 153,164,171,172,175,188,207,212,236,242,243,244,245,261,268,275,279,
%U A030515 284,292,316,325,332,333,338,356,363,369,387,388,404,412,423,425,428
%N A030515 Numbers with exactly 6 divisors.
%C A030515 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
%C A030515 Also numbers which are the square root of the product of their proper divisors. - _Amarnath Murthy_, Apr 21 2001
%C A030515 Such numbers are multiplicatively 3-perfect (i.e., the product of divisors of a(n) equals a(n)^3). - _Lekraj Beedassy_, Jul 13 2005
%C A030515 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
%D A030515 Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%H A030515 R. J. Mathar, <a href="/A030515/b030515.txt">Table of n, a(n) for n = 1..1000</a>
%H A030515 Amarnath Murthy and Charles Ashbacher, <a href="http://www.gallup.unm.edu/~smarandache/MurthyBook.pdf">Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences</a>, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.
%H A030515 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>
%F A030515 Union of A050997 and A054753. - _Lekraj Beedassy_, Jul 13 2005
%F A030515 A000005(a(n))=6. - _Juri-Stepan Gerasimov_, Oct 10 2009
%p A030515 N:= 1000: # to get all terms <= N
%p A030515 Primes:= select(isprime, {2,seq(i,i=3..floor(N/4))}):
%p A030515 S:= select(`<=`,{seq(p^5, p = Primes),seq(seq(p*q^2, p=Primes minus {q}),q=Primes)},N):
%p A030515 sort(convert(S,list)); # _Robert Israel_, Feb 10 2016
%t A030515 f[n_]:=Length[Divisors[n]]==6; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 14 2009 *)
%t A030515 Select[Range[500],DivisorSigma[0,#]==6&] (* _Harvey P. Dale_, Oct 02 2014 *)
%o A030515 (PARI) is(n)=numdiv(n)==6 \\ _Charles R Greathouse IV_, Jan 23 2014
%o A030515 (Python)
%o A030515 from sympy import divisor_count
%o A030515 def ok(n): return divisor_count(n) == 6
%o A030515 print([k for k in range(429) if ok(k)]) # _Michael S. Branicky_, Dec 18 2021
%o A030515 (Python)
%o A030515 from math import isqrt
%o A030515 from sympy import primepi, primerange, integer_nthroot
%o A030515 def A030515(n):
%o A030515     def bisection(f,kmin=0,kmax=1):
%o A030515         while f(kmax) > kmax: kmax <<= 1
%o A030515         kmin = kmax >> 1
%o A030515         while kmax-kmin > 1:
%o A030515             kmid = kmax+kmin>>1
%o A030515             if f(kmid) <= kmid:
%o A030515                 kmax = kmid
%o A030515             else:
%o A030515                 kmin = kmid
%o A030515         return kmax
%o A030515     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])
%o A030515     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A030515 Cf. A061117.
%K A030515 nonn,easy
%O A030515 1,1
%A A030515 _Jeff Burch_
%E A030515 Definition clarified by _Jonathan Sondow_, Jan 23 2014