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A072774 Powers of squarefree numbers.

%I A072774 #50 Jun 10 2025 17:18:11
%S A072774 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,
%T A072774 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,
%U A072774 65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94,95,97
%N A072774 Powers of squarefree numbers.
%C A072774 Essentially the same as A062770. - _R. J. Mathar_, Sep 25 2008
%C A072774 Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - _Reinhard Zumkeller_, Apr 06 2014
%C A072774 Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - _Gus Wiseman_, Apr 16 2018
%H A072774 Reinhard Zumkeller, <a href="/A072774/b072774.txt">Table of n, a(n) for n = 1..10000</a>
%F A072774 a(n) = A072775(n)^A072776(n).
%F A072774 Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - _Amiram Eldar_, Mar 20 2025
%F A072774 a(n)/n ~ Pi^2/6 (A013661). - _Friedjof Tellkamp_, Jun 09 2025
%p A072774 isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
%p A072774 select(isA, [seq(1..97)]);  # _Peter Luschny_, Jun 10 2025
%t A072774 Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* _Geoffrey Critzer_, Mar 30 2015 *)
%o A072774 (Haskell)
%o A072774 import Data.Map (empty, findMin, deleteMin, insert)
%o A072774 import qualified Data.Map.Lazy as Map (null)
%o A072774 a072774 n = a072774_list !! (n-1)
%o A072774 (a072774_list, a072775_list, a072776_list) = unzip3 $
%o A072774    (1, 1, 1) : f (tail a005117_list) empty where
%o A072774    f vs'@(v:vs) m
%o A072774     | Map.null m || xx > v = (v, v, 1) :
%o A072774                              f vs (insert (v^2) (v, 2) m)
%o A072774     | otherwise = (xx, bx, ex) :
%o A072774                   f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
%o A072774     where (xx, (bx, ex)) = findMin m
%o A072774 -- _Reinhard Zumkeller_, Apr 06 2014
%o A072774 (PARI) is(n)=ispower(n,,&n); issquarefree(n) \\ _Charles R Greathouse IV_, Oct 16 2015
%o A072774 (Python)
%o A072774 from math import isqrt
%o A072774 from sympy import mobius, integer_nthroot
%o A072774 def A072774(n):
%o A072774     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
%o A072774     def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))
%o A072774     kmin, kmax = 1,2
%o A072774     while f(kmax) >= kmax:
%o A072774         kmax <<= 1
%o A072774     while True:
%o A072774         kmid = kmax+kmin>>1
%o A072774         if f(kmid) < kmid:
%o A072774             kmax = kmid
%o A072774         else:
%o A072774             kmin = kmid
%o A072774         if kmax-kmin <= 1:
%o A072774             break
%o A072774     return kmax # _Chai Wah Wu_, Aug 19 2024
%Y A072774 Complement of A059404.
%Y A072774 Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
%Y A072774 A subsequence of A242414.
%Y A072774 Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.
%K A072774 nonn
%O A072774 1,2
%A A072774 _Reinhard Zumkeller_, Jul 10 2002