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A069492 5-full numbers: if a prime p divides k then so does p^5.

%I A069492 #28 Sep 11 2024 00:32:23
%S A069492 1,32,64,128,243,256,512,729,1024,2048,2187,3125,4096,6561,7776,8192,
%T A069492 15552,15625,16384,16807,19683,23328,31104,32768,46656,59049,62208,
%U A069492 65536,69984,78125,93312,100000,117649,124416,131072,139968,161051
%N A069492 5-full numbers: if a prime p divides k then so does p^5.
%C A069492 a(m) mod prime(n) > 0 for m < A258602(n); a(A258602(n)) = A050997(n) = prime(n)^5. - _Reinhard Zumkeller_, Jun 06 2015
%H A069492 Reinhard Zumkeller, <a href="/A069492/b069492.txt">Table of n, a(n) for n = 1..10000</a> (terms n=148..10000 corrected by Andrew Howroyd)
%F A069492 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^4*(p-1))) = 1.0695724994489739263413712783666538355049945684326048537289707764272637... - _Amiram Eldar_, Jul 09 2020
%o A069492 (PARI) for(n=1,250000,if(n*sumdiv(n,d,isprime(d)/d^5)==floor(n*sumdiv(n,d,isprime(d)/d^5)),print1(n,",")))
%o A069492 (PARI) \\ Gen(limit,k) defined in A036967.
%o A069492 Gen(170000, 5) \\ _Andrew Howroyd_, Sep 10 2024
%o A069492 (Haskell)
%o A069492 import Data.Set (singleton, deleteFindMin, fromList, union)
%o A069492 a069492 n = a069492_list !! (n-1)
%o A069492 a069492_list = 1 : f (singleton z) [1, z] zs where
%o A069492    f s q5s p5s'@(p5:p5s)
%o A069492      | m < p5 = m : f (union (fromList $ map (* m) ps) s') q5s p5s'
%o A069492      | otherwise = f (union (fromList $ map (* p5) q5s) s) (p5:q5s) p5s
%o A069492      where ps = a027748_row m
%o A069492            (m, s') = deleteFindMin s
%o A069492    (z:zs) = a050997_list
%o A069492 -- _Reinhard Zumkeller_, Jun 03 2015
%o A069492 (Python)
%o A069492 from math import gcd
%o A069492 from sympy import integer_nthroot, factorint
%o A069492 def A069492(n):
%o A069492     def bisection(f,kmin=0,kmax=1):
%o A069492         while f(kmax) > kmax: kmax <<= 1
%o A069492         while kmax-kmin > 1:
%o A069492             kmid = kmax+kmin>>1
%o A069492             if f(kmid) <= kmid:
%o A069492                 kmax = kmid
%o A069492             else:
%o A069492                 kmin = kmid
%o A069492         return kmax
%o A069492     def f(x):
%o A069492         c = n+x
%o A069492         for t in range(1,integer_nthroot(x,9)[0]+1):
%o A069492             if all(d<=1 for d in factorint(t).values()):
%o A069492                 for u in range(1,integer_nthroot(s:=x//t**9,8)[0]+1):
%o A069492                     if gcd(t,u)==1 and all(d<=1 for d in factorint(u).values()):
%o A069492                         for w in range(1,integer_nthroot(a:=s//u**8,7)[0]+1):
%o A069492                             if gcd(u,w)==1 and gcd(t,w)==1 and all(d<=1 for d in factorint(w).values()):
%o A069492                                 for y in range(1,integer_nthroot(z:=a//w**7,6)[0]+1):
%o A069492                                     if gcd(w,y)==1 and gcd(u,y)==1 and gcd(t,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A069492                                         c -= integer_nthroot(z//y**6,5)[0]
%o A069492         return c
%o A069492     return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A069492 Cf. A036967, A036966, A001694.
%Y A069492 Cf. A050997.
%Y A069492 Cf. A258602.
%K A069492 easy,nonn
%O A069492 1,2
%A A069492 _Benoit Cloitre_, Apr 15 2002