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A189883 Numbers k such that the square part of k is one greater than the squarefree part of k.

%I A189883 #19 Jun 28 2025 19:26:59
%S A189883 12,240,1260,20592,38220,65280,104652,159600,233772,809100,1047552,
%T A189883 1335180,1678320,2083692,2558400,3109932,7308912,8500140,9831360,
%U A189883 11313132,12956400,18970380,21376752,24005100,26868672,37008972,49780080
%N A189883 Numbers k such that the square part of k is one greater than the squarefree part of k.
%C A189883 The complementary sequence, squarefree part of k is one greater than the square part of k, is A069187.
%H A189883 Alois P. Heinz, <a href="/A189883/b189883.txt">Table of n, a(n) for n = 1..1000</a>
%H A189883 Antonio Roldán, <a href="http://hojaynumeros.blogspot.com">hojaynumeros.blogspot.com</a>
%F A189883 n such that A008833(n) - A007913(n) = 1.
%F A189883 a(n) = x^2 (x^2-1), where x = A067874(n). - _T. D. Noe_, Apr 29 2011
%e A189883 1260 = 2^2*3^2*5*7, square part: 2^2*3^2 = 36, squarefree part: 5*7 = 35, and 36 = 35+1.
%p A189883 b:= proc() 1 end:
%p A189883 a:= proc(n) option remember; local i, k;
%p A189883       if n>1 then a(n-1) fi;
%p A189883       for k from b(n-1)+1 while 1<>mul(i[2], i=ifactors(k^2-1)[2])
%p A189883       do od; b(n):=k; k^4-k^2
%p A189883     end:
%p A189883 seq(a(n), n=1..50); # _Alois P. Heinz_, Apr 29 2011
%t A189883 okQ[n_] := Module[{p, e, sfp}, {p, e} = Transpose[FactorInteger[n]]; e = Mod[e, 2]; sfp = Times @@ (p^e); n/sfp - sfp == 1]; Select[Range[10^5], okQ] (* _T. D. Noe_, Apr 29 2011 *)
%o A189883 (PARI) for(n=1,1e3,if(issquarefree(n^2-1),print1(n^4-n^2", "))) \\ _Charles R Greathouse IV_, Apr 29 2011
%K A189883 nonn
%O A189883 1,1
%A A189883 _Antonio Roldán_, Apr 29 2011