A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.
1, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 432, 512, 625, 648, 729, 864, 1000, 1024, 1296, 1331, 1728, 1944, 2000, 2048, 2187, 2197, 2401, 2592, 2744, 3125, 3375, 3456, 3888, 4000, 4096, 4913, 5000, 5184, 5488, 5832, 6561, 6859, 6912, 7776, 8000
Offset: 1
References
- M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
- A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
- E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
- P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
- M. J. Halm, Sequences
- H.-Q. Liu, The number of cubefull numbers in an interval (supplement), Funct. Approx. Comment. Math. 43 (2) 105-107, December 2010.
- Index entries for sequences related to powerful numbers
Programs
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Haskell
import Data.Set (singleton, deleteFindMin, fromList, union) a036966 n = a036966_list !! (n-1) a036966_list = 1 : f (singleton z) [1, z] zs where f s q3s p3s'@(p3:p3s) | m < p3 = m : f (union (fromList $ map (* m) ps) s') q3s p3s' | otherwise = f (union (fromList $ map (* p3) q3s) s) (p3:q3s) p3s where ps = a027748_row m (m, s') = deleteFindMin s (z:zs) = a030078_list -- Reinhard Zumkeller, Jun 03 2015, Dec 15 2013 -
Maple
isA036966 := proc(n) local p ; for p in ifactors(n)[2] do if op(2,p) < 3 then return false; end if; end do: return true ; end proc: A036966 := proc(n) option remember; if n = 1 then 1 ; else for a from procname(n-1)+1 do if isA036966(a) then return a; end if; end do: end if; end proc: # R. J. Mathar, May 01 2013
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Mathematica
Select[ Range[2, 8191], Min[ Table[ # [[2]], {1}] & /@ FactorInteger[ # ]] > 2 &] Join[{1},Select[Range[8000],Min[Transpose[FactorInteger[#]][[2]]]>2&]] (* Harvey P. Dale, Jul 17 2013 *) -
PARI
is(n)=n==1 || vecmin(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015
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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
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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
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Python
from math import gcd from sympy import integer_nthroot, factorint def A036966(n): def f(x): c = n+x for w in range(1,integer_nthroot(x,5)[0]+1): if all(d<=1 for d in factorint(w).values()): for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1): if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()): c -= integer_nthroot(z//y**4,3)[0] return c def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024
Formula
Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) (A065483). - Amiram Eldar, Jun 23 2020
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
Extensions
More terms from Erich Friedman
Corrected by Vladeta Jovovic, Aug 17 2002
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