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%I #56 Apr 18 2023 10:20:40
%S 1,8,16,27,32,64,81,125,128,216,243,256,343,432,512,625,648,729,864,
%T 1000,1024,1296,1331,1728,1944,2000,2048,2187,2197,2401,2592,2744,
%U 3125,3375,3456,3888,4000,4096,4913,5000,5184,5488,5832,6561,6859,6912,7776,8000
%N 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.
%C Also called powerful_3 numbers.
%C For n > 1: A124010(a(n),k) > 2, k = 1..A001221(a(n)). - _Reinhard Zumkeller_, Dec 15 2013
%C a(m) mod prime(n) > 0 for m < A258600(n); a(A258600(n)) = A030078(n) = prime(n)^3. - _Reinhard Zumkeller_, Jun 06 2015
%D M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
%D A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
%D E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.
%H Amiram Eldar, <a href="/A036966/b036966.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H P. Erdős and G. Szekeres, <a href="http://www.renyi.hu/~p_erdos/1934-05.pdf">Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem</a>, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
%H M. J. Halm, <a href="http://untilheaven.tripod.com/id112.htm">Sequences</a>
%H H.-Q. Liu, <a href="https://doi.org/10.7169/facm/1291903391">The number of cubefull numbers in an interval (supplement)</a>, Funct. Approx. Comment. Math. 43 (2) 105-107, December 2010.
%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>
%F Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) (A065483). - _Amiram Eldar_, Jun 23 2020
%F Numbers of the form x^5*y^4*z^3. There is a unique representation with x,y squarefree and coprime. - _Charles R Greathouse IV_, Jan 12 2022
%p isA036966 := proc(n)
%p local p ;
%p for p in ifactors(n)[2] do
%p if op(2,p) < 3 then
%p return false;
%p end if;
%p end do:
%p return true ;
%p end proc:
%p A036966 := proc(n)
%p option remember;
%p if n = 1 then
%p 1 ;
%p else
%p for a from procname(n-1)+1 do
%p if isA036966(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, May 01 2013
%t Select[ Range[2, 8191], Min[ Table[ # [[2]], {1}] & /@ FactorInteger[ # ]] > 2 &]
%t Join[{1},Select[Range[8000],Min[Transpose[FactorInteger[#]][[2]]]>2&]] (* _Harvey P. Dale_, Jul 17 2013 *)
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, fromList, union)
%o a036966 n = a036966_list !! (n-1)
%o a036966_list = 1 : f (singleton z) [1, z] zs where
%o f s q3s p3s'@(p3:p3s)
%o | m < p3 = m : f (union (fromList $ map (* m) ps) s') q3s p3s'
%o | otherwise = f (union (fromList $ map (* p3) q3s) s) (p3:q3s) p3s
%o where ps = a027748_row m
%o (m, s') = deleteFindMin s
%o (z:zs) = a030078_list
%o -- _Reinhard Zumkeller_, Jun 03 2015, Dec 15 2013
%o (PARI) is(n)=n==1 || vecmin(factor(n)[,2])>2 \\ _Charles R Greathouse IV_, Sep 17 2015
%o (PARI) list(lim)=my(v=List(),t); for(a=1,sqrtnint(lim\1,5), for(b=1,sqrtnint(lim\a^5,4), t=a^5*b^4; for(c=1,sqrtnint(lim\t,3), listput(v,t*c^3)))); Set(v) \\ _Charles R Greathouse IV_, Nov 20 2015
%o (PARI) list(lim)=my(v=List(),t); forsquarefree(a=1,sqrtnint(lim\1,5), my(a5=a[1]^5); forsquarefree(b=1,sqrtnint(lim\a5,4), if(gcd(a[1],b[1])>1, next); t=a5*b[1]^4; for(c=1,sqrtnint(lim\t,3), listput(v,t*c^3)))); Set(v) \\ _Charles R Greathouse IV_, Jan 12 2022
%Y Cf. A001694, A030078, A036967, A065483, A258600.
%K easy,nonn,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Erich Friedman_
%E Corrected by _Vladeta Jovovic_, Aug 17 2002
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