A062320 - OEIS

%I #30 Sep 18 2024 11:21:50

%S 16,64,81,144,256,324,400,576,625,729,784,1024,1296,1600,1936,2025,

%T 2304,2401,2500,2704,2916,3136,3600,3969,4096,4624,5184,5625,5776,

%U 6400,6561,7056,7744,8100,8464,9216,9604,9801,10000,10816,11664,12544,13456

%N Nonsquarefree numbers squared. A013929(n)^2.

%C A008966(A037213(a(n))) = 0. - _Reinhard Zumkeller_, Sep 03 2015

%H Harry J. Smith, <a href="/A062320/b062320.txt">Table of n, a(n) for n = 1..1000</a>

%F Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2. - _Amiram Eldar_, Jul 16 2020

%o (PARI) for(n=1,55, if(issquarefree(n), n+1,print(n^2)))

%o (PARI) n=-1; for (m=1, 10^9, if (!issquarefree(m), write("b062320.txt", n++, " ", m^2); if (n==1000, break))) \\ _Harry J. Smith_, Aug 04 2009

%o (PARI) is(n)=issquare(n,&n) && !issquarefree(n) \\ _Charles R Greathouse IV_, Sep 18 2015

%o (Haskell)

%o a062320 = (^ 2) . a013929 -- _Reinhard Zumkeller_, Sep 03 2015

%o (Python)

%o from math import isqrt

%o from sympy import mobius

%o def A062320(n):

%o     def f(x): return n+1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     return bisection(f)**2 # _Chai Wah Wu_, Aug 31 2024

%Y Cf. A013929, A008966, A037213, A320965.

%Y Squares in A046099.

%K easy,nonn

%O 1,1

%A _Jason Earls_, Jul 05 2001

%E More terms from Larry Reeves (larryr(AT)acm.org), Jul 11 2001

%E Offset corrected by _Andrew Howroyd_, Sep 18 2024