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%I #53 May 05 2023 09:51:17
%S 12,18,20,28,32,44,45,50,52,63,68,75,76,92,98,99,116,117,124,147,148,
%T 153,164,171,172,175,188,207,212,236,242,243,244,245,261,268,275,279,
%U 284,292,316,325,332,333,338,356,363,369,387,388,404,412,423,425,428
%N Numbers with exactly 6 divisors.
%C 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 Also numbers which are the square root of the product of their proper divisors. - _Amarnath Murthy_, Apr 21 2001
%C 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 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 Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%H R. J. Mathar, <a href="/A030515/b030515.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>
%F Union of A050997 and A054753. - _Lekraj Beedassy_, Jul 13 2005
%F A000005(a(n))=6. - _Juri-Stepan Gerasimov_, Oct 10 2009
%p N:= 1000: # to get all terms <= N
%p Primes:= select(isprime, {2,seq(i,i=3..floor(N/4))}):
%p S:= select(`<=`,{seq(p^5, p = Primes),seq(seq(p*q^2, p=Primes minus {q}),q=Primes)},N):
%p sort(convert(S,list)); # _Robert Israel_, Feb 10 2016
%t 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 Select[Range[500],DivisorSigma[0,#]==6&] (* _Harvey P. Dale_, Oct 02 2014 *)
%o (PARI) is(n)=numdiv(n)==6 \\ _Charles R Greathouse IV_, Jan 23 2014
%o (Python)
%o from sympy import divisor_count
%o def ok(n): return divisor_count(n) == 6
%o print([k for k in range(429) if ok(k)]) # _Michael S. Branicky_, Dec 18 2021
%Y Cf. A061117.
%K nonn,easy
%O 1,1
%A _Jeff Burch_
%E Definition clarified by _Jonathan Sondow_, Jan 23 2014
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