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A069493 6-full numbers: if p divides n then so does p^6.

%I A069493 #25 Sep 10 2024 08:44:43
%S A069493 1,64,128,256,512,729,1024,2048,2187,4096,6561,8192,15625,16384,19683,
%T A069493 32768,46656,59049,65536,78125,93312,117649,131072,139968,177147,
%U A069493 186624,262144,279936,373248,390625,419904,524288,531441,559872,746496,823543,839808
%N A069493 6-full numbers: if p divides n then so does p^6.
%C A069493 a(m) mod prime(n) > 0 for m < A258603(n); a(A258600(n)) = A030516(n) = prime(n)^6. - _Reinhard Zumkeller_, Jun 06 2015
%H A069493 Alois P. Heinz, <a href="/A069493/b069493.txt">Table of n, a(n) for n = 1..10000</a>
%F A069493 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^5*(p-1))) = 1.0334657852594050612296726462481884631303137561267151463866539131591664... - _Amiram Eldar_, Jul 09 2020
%e A069493 2^7*3^6 = 93312 is a member (although not of A076470).
%t A069493 Join[{1},Select[Range[900000],Min[FactorInteger[#][[All,2]]]>5&]] (* _Harvey P. Dale_, Mar 03 2018 *)
%o A069493 (PARI) for(n=1,560000,if(n*sumdiv(n,d,isprime(d)/d^6)==floor(n*sumdiv(n,d,isprime(d)/d^6)),print1(n,",")))
%o A069493 (Haskell)
%o A069493 import Data.Set (singleton, deleteFindMin, fromList, union)
%o A069493 a069493 n = a069493_list !! (n-1)
%o A069493 a069493_list = 1 : f (singleton z) [1, z] zs where
%o A069493    f s q6s p6s'@(p6:p6s)
%o A069493      | m < p6 = m : f (union (fromList $ map (* m) ps) s') q6s p6s'
%o A069493      | otherwise = f (union (fromList $ map (* p6) q6s) s) (p6:q6s) p6s
%o A069493      where ps = a027748_row m
%o A069493            (m, s') = deleteFindMin s
%o A069493    (z:zs) = a030516_list
%o A069493 -- _Reinhard Zumkeller_, Jun 03 2015
%o A069493 (Python)
%o A069493 from math import gcd
%o A069493 from sympy import factorint, integer_nthroot
%o A069493 def A069493(n):
%o A069493     def f(x):
%o A069493         c = n+x
%o A069493         for y1 in range(1,integer_nthroot(x,11)[0]+1):
%o A069493             if all(d<=1 for d in factorint(y1).values()):
%o A069493                 for y2 in range(1,integer_nthroot(z2:=x//y1**11,10)[0]+1):
%o A069493                     if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
%o A069493                         for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
%o A069493                             if gcd(y3,y1)==1 and gcd(y3,y2)==1 and all(d<=1 for d in factorint(y3).values()):
%o A069493                                 for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
%o A069493                                     if gcd(y4,y1)==1 and gcd(y4,y2)==1 and gcd(y4,y3)==1 and all(d<=1 for d in factorint(y4).values()):
%o A069493                                         for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
%o A069493                                             if gcd(y5,y1)==1 and gcd(y5,y2)==1 and gcd(y5,y3)==1 and gcd(y5,y4)==1 and all(d<=1 for d in factorint(y5).values()):
%o A069493                                                 c -= integer_nthroot(z5//y5**7,6)[0]
%o A069493         return c
%o A069493     def bisection(f,kmin=0,kmax=1):
%o A069493         while f(kmax) > kmax: kmax <<= 1
%o A069493         while kmax-kmin > 1:
%o A069493             kmid = kmax+kmin>>1
%o A069493             if f(kmid) <= kmid:
%o A069493                 kmax = kmid
%o A069493             else:
%o A069493                 kmin = kmid
%o A069493         return kmax
%o A069493     return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A069493 Cf. A036967, A036966, A001694. Different from A076470.
%Y A069493 Cf. A030516, A258603.
%K A069493 easy,nonn
%O A069493 1,2
%A A069493 _Benoit Cloitre_, Apr 15 2002