A036967 4-full numbers: if a prime p divides k then so does p^4.

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 2592, 3125, 3888, 4096, 5184, 6561, 7776, 8192, 10000, 10368, 11664, 14641, 15552, 15625, 16384, 16807, 19683, 20000, 20736, 23328, 28561, 31104, 32768, 34992

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

a(m) mod prime(n) > 0 for m < A258601(n); a(A258601(n)) = A030514(n) = prime(n)^4. - Reinhard Zumkeller, Jun 06 2015

References

• E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 300 terms from T. D. Noe)

Crossrefs

A030514 is a subsequence.

Cf. A001694, A036966, A046101, A258601.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, fromList, union)
a036967 n = a036967_list !! (n-1)
a036967_list = 1 : f (singleton z) [1, z] zs where
   f s q4s p4s'@(p4:p4s)
     | m < p4 = m : f (union (fromList $ map (* m) ps) s') q4s p4s'
     | otherwise = f (union (fromList $ map (* p4) q4s) s) (p4:q4s) p4s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030514_list
-- Reinhard Zumkeller, Jun 03 2015

Mathematica

Join[{1},Select[Range[35000],Min[Transpose[FactorInteger[#]][[2]]]>3&]] (* Harvey P. Dale, Jun 05 2012 *)

PARI

is(n)=n==1 || vecmin(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 17 2015

PARI

M(v,u,lim)=vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))
Gen(lim,k)={my(v=[1]); forprime(p=2, sqrtnint(lim, k), v=M(v, concat([1], vector(logint(lim,p)-k+1,e,p^(e+k-1))), lim));v}
Gen(35000,4) \\ Andrew Howroyd, Sep 10 2024

Python

from sympy import factorint
A036967_list = [n for n in range(1,10**5) if min(factorint(n).values(),default=4) >= 4] # Chai Wah Wu, Aug 18 2021

Python

from math import gcd
from sympy import integer_nthroot, factorint
def A036967(n):
    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
    def f(x):
        c = n+x
        for u in range(1,integer_nthroot(x,7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):
                    if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                            if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                                c -= integer_nthroot(z//y**5,4)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.1488462139214317030108176090790939019972506733993367867997411290952527... - Amiram Eldar, Jul 09 2020

Extensions

More terms from Erich Friedman

Corrected by Vladeta Jovovic, Aug 17 2002