cp's OEIS Frontend

A062503 Squarefree numbers squared.

%I A062503 #97 Aug 19 2024 13:16:48
%S A062503 1,4,9,25,36,49,100,121,169,196,225,289,361,441,484,529,676,841,900,
%T A062503 961,1089,1156,1225,1369,1444,1521,1681,1764,1849,2116,2209,2601,2809,
%U A062503 3025,3249,3364,3481,3721,3844,4225,4356,4489,4761,4900,5041,5329,5476
%N A062503 Squarefree numbers squared.
%C A062503 Also, except for the initial term, numbers whose prime factors are squared. - _Cino Hilliard_, Jan 25 2006
%C A062503 Also cubefree numbers that are squares. - _Gionata Neri_, May 08 2016
%C A062503 All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (term of this sequence), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on. - _Peter Munn_, Mar 12 2020
%C A062503 Powerful numbers (A001694) all of whose nonunitary divisors are not powerful (A052485). - _Amiram Eldar_, May 13 2023
%H A062503 Harry J. Smith, <a href="/A062503/b062503.txt">Table of n, a(n) for n = 1..1000</a>
%F A062503 Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - _Benoit Cloitre_, Mar 03 2004
%F A062503 a(n) = A000290(A005117(n)); A227291(a(n)) = 1. - _Reinhard Zumkeller_, Jul 07 2013
%F A062503 A000290 \ A062320. - _R. J. Mathar_, Jul 27 2013
%F A062503 a(n) ~ (Pi^4/36) * n^2. - _Charles R Greathouse IV_, Nov 24 2015
%F A062503 a(n) = A046692(a(n))^2. - _Torlach Rush_, Jan 05 2019
%F A062503 For all k in the sequence, Omega(k) = 2*omega(k). - _Wesley Ivan Hurt_, Apr 30 2020
%F A062503 Sum_{n>=1} 1/a(n) = zeta(2)/zeta(4) = 15/Pi^2 (A082020). - _Amiram Eldar_, May 22 2020
%t A062503 Select[Range[100], SquareFreeQ]^2
%o A062503 (PARI) je=[]; for(n=1,200, if(issquarefree(n),je=concat(je,n^2),)); je
%o A062503 (PARI) n=0; for (m=1, 10^5, if(issquarefree(m), write("b062503.txt", n++, " ", m^2); if (n==1000, break))) \\ _Harry J. Smith_, Aug 08 2009
%o A062503 (PARI) is(n)=issquare(n,&n) && issquarefree(n) \\ _Charles R Greathouse IV_, Sep 18 2015
%o A062503 (Haskell)
%o A062503 a062503 = a000290 . a005117  -- _Reinhard Zumkeller_, Jul 07 2013
%o A062503 (Python)
%o A062503 from math import isqrt
%o A062503 from sympy import mobius
%o A062503 def A062503(n):
%o A062503     def f(x): return n-1+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A062503     kmin, kmax = 1,2
%o A062503     while f(kmax) >= kmax:
%o A062503         kmax <<= 1
%o A062503     while True:
%o A062503         kmid = kmax+kmin>>1
%o A062503         if f(kmid) < kmid:
%o A062503             kmax = kmid
%o A062503         else:
%o A062503             kmin = kmid
%o A062503         if kmax-kmin <= 1:
%o A062503             break
%o A062503     return kmax**2 # _Chai Wah Wu_, Aug 19 2024
%Y A062503 Characteristic function is A227291.
%Y A062503 Other powers of squarefree numbers: A005117(1), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7), A072774(all).
%Y A062503 Cf. A000290, A001694, A052485, A062320.
%Y A062503 Cf. A001248 (a subsequence).
%Y A062503 A329332 column 2 in ascending order.
%K A062503 nonn
%O A062503 1,2
%A A062503 _Jason Earls_, Jul 09 2001