cp's OEIS Frontend

A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

%I A051144 #91 Apr 21 2025 23:05:17
%S A051144 8,12,18,20,24,27,28,32,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,
%T A051144 84,88,90,92,96,98,99,104,108,112,116,117,120,124,125,126,128,132,135,
%U A051144 136,140,147,148,150,152,153,156,160,162,164,168,171,172,175,176,180
%N A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.
%C A051144 At least one exponent in the canonical prime factorization (cf. A124010) of n is odd, and at least one exponent is greater than 1. - _Reinhard Zumkeller_, Jan 24 2013
%C A051144 Compare this sequence, as a set, with A177712, numbers that have an odd factor, but are not odd. The self-inverse function defined by A225546, maps the members of either one of these sets 1:1 onto the other set. - _Peter Munn_, Jul 31 2020
%H A051144 Reinhard Zumkeller, <a href="/A051144/b051144.txt">Table of n, a(n) for n = 1..10000</a>
%H A051144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.
%H A051144 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%H A051144 Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_number">Square number</a>.
%H A051144 Wikipedia, <a href="http://en.wikipedia.org/wiki/Squarefree_integer">Squarefree integer</a>.
%F A051144 (1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - _Reinhard Zumkeller_, Jan 24 2013
%F A051144 Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(2*s) - zeta(s)/zeta(2*s), for s > 1. - _Amiram Eldar_, Dec 03 2022
%e A051144 63 is included because 63 = 3^2 * 7.
%e A051144 64 is not included because it is a perfect square (8^2).
%e A051144 65 is not included because it is squarefree (5 * 13).
%p A051144 N:= 10000;  # to get all entries up to N
%p A051144 A051144:= remove(numtheory:-issqrfree,{$1..N}) minus {seq(i^2,i=1..floor(sqrt(N)))}:
%p A051144 # _Robert Israel_, Mar 30 2014
%t A051144 searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* _Alonso del Arte_, Dec 20 2019 *)
%o A051144 (Haskell)
%o A051144 a051144 n = a051144_list !! (n-1)
%o A051144 a051144_list = filter ((== 0) . a008966) a000037_list
%o A051144 -- _Reinhard Zumkeller_, Sep 02 2013, Jan 24 2013
%o A051144 (PARI) is(n)=!issquare(n) && !issquarefree(n) \\ _Charles R Greathouse IV_, Sep 18 2015
%o A051144 (Magma) [k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // _Marius A. Burtea_, Dec 29 2019
%o A051144 (Python)
%o A051144 from math import isqrt
%o A051144 from sympy import mobius
%o A051144 def A051144(n):
%o A051144     def bisection(f,kmin=0,kmax=1):
%o A051144         while f(kmax) > kmax: kmax <<= 1
%o A051144         kmin = kmax >> 1
%o A051144         while kmax-kmin > 1:
%o A051144             kmid = kmax+kmin>>1
%o A051144             if f(kmid) <= kmid:
%o A051144                 kmax = kmid
%o A051144             else:
%o A051144                 kmin = kmid
%o A051144         return kmax
%o A051144     def f(x): return int(n-1+(y:=isqrt(x))+sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
%o A051144     return bisection(f,n,n) # _Chai Wah Wu_, Mar 23 2025
%Y A051144 Cf. A210490 (complement), intersection of A013929 and A000037.
%Y A051144 Related to A177712 via A225546.
%K A051144 nonn,easy
%O A051144 1,1
%A A051144 Michael Minic (Rassilon6(AT)aol.com)
%E A051144 Incorrect comment removed by _Charles R Greathouse IV_, Mar 19 2010
%E A051144 Offset corrected by _Reinhard Zumkeller_, Jan 24 2013