cp's OEIS Frontend

A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.

%I A036966 #65 Sep 22 2024 03:55:20
%S A036966 1,8,16,27,32,64,81,125,128,216,243,256,343,432,512,625,648,729,864,
%T A036966 1000,1024,1296,1331,1728,1944,2000,2048,2187,2197,2401,2592,2744,
%U A036966 3125,3375,3456,3888,4000,4096,4913,5000,5184,5488,5832,6561,6859,6912,7776,8000
%N A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.
%C A036966 Also called powerful_3 numbers.
%C A036966 For n > 1: A124010(a(n),k) > 2, k = 1..A001221(a(n)). - _Reinhard Zumkeller_, Dec 15 2013
%C A036966 a(m) mod prime(n) > 0 for m < A258600(n); a(A258600(n)) = A030078(n) = prime(n)^3. - _Reinhard Zumkeller_, Jun 06 2015
%D A036966 M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
%D A036966 A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
%D A036966 E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.
%H A036966 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 A036966 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 A036966 M. J. Halm, <a href="http://untilheaven.tripod.com/id112.htm">Sequences</a>
%H A036966 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 A036966 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>
%F A036966 Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) (A065483). - _Amiram Eldar_, Jun 23 2020
%F A036966 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 A036966 isA036966 := proc(n)
%p A036966     local p ;
%p A036966     for p in ifactors(n)[2] do
%p A036966         if op(2,p) < 3 then
%p A036966             return false;
%p A036966         end if;
%p A036966     end do:
%p A036966     return true ;
%p A036966 end proc:
%p A036966 A036966 := proc(n)
%p A036966     option remember;
%p A036966     if n = 1 then
%p A036966         1 ;
%p A036966     else
%p A036966         for a from procname(n-1)+1 do
%p A036966             if isA036966(a) then
%p A036966                 return a;
%p A036966             end if;
%p A036966         end do:
%p A036966     end if;
%p A036966 end proc: # _R. J. Mathar_, May 01 2013
%t A036966 Select[ Range[2, 8191], Min[ Table[ # [[2]], {1}] & /@ FactorInteger[ # ]] > 2 &]
%t A036966 Join[{1},Select[Range[8000],Min[Transpose[FactorInteger[#]][[2]]]>2&]] (* _Harvey P. Dale_, Jul 17 2013 *)
%o A036966 (Haskell)
%o A036966 import Data.Set (singleton, deleteFindMin, fromList, union)
%o A036966 a036966 n = a036966_list !! (n-1)
%o A036966 a036966_list = 1 : f (singleton z) [1, z] zs where
%o A036966    f s q3s p3s'@(p3:p3s)
%o A036966      | m < p3 = m : f (union (fromList $ map (* m) ps) s') q3s p3s'
%o A036966      | otherwise = f (union (fromList $ map (* p3) q3s) s) (p3:q3s) p3s
%o A036966      where ps = a027748_row m
%o A036966            (m, s') = deleteFindMin s
%o A036966    (z:zs) = a030078_list
%o A036966 -- _Reinhard Zumkeller_, Jun 03 2015, Dec 15 2013
%o A036966 (PARI) is(n)=n==1 || vecmin(factor(n)[,2])>2 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A036966 (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 A036966 (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
%o A036966 (Python)
%o A036966 from math import gcd
%o A036966 from sympy import integer_nthroot, factorint
%o A036966 def A036966(n):
%o A036966     def f(x):
%o A036966         c = n+x
%o A036966         for w in range(1,integer_nthroot(x,5)[0]+1):
%o A036966             if all(d<=1 for d in factorint(w).values()):
%o A036966                 for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
%o A036966                     if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A036966                         c -= integer_nthroot(z//y**4,3)[0]
%o A036966         return c
%o A036966     def bisection(f,kmin=0,kmax=1):
%o A036966         while f(kmax) > kmax: kmax <<= 1
%o A036966         while kmax-kmin > 1:
%o A036966             kmid = kmax+kmin>>1
%o A036966             if f(kmid) <= kmid:
%o A036966                 kmax = kmid
%o A036966             else:
%o A036966                 kmin = kmid
%o A036966         return kmax
%o A036966     return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A036966 Cf. A001694, A030078, A036967, A065483, A258600, A376363, A376364.
%K A036966 easy,nonn,nice
%O A036966 1,2
%A A036966 _N. J. A. Sloane_
%E A036966 More terms from _Erich Friedman_
%E A036966 Corrected by _Vladeta Jovovic_, Aug 17 2002