A189883 - OEIS

%I #16 Nov 14 2014 11:12:31

%S 12,240,1260,20592,38220,65280,104652,159600,233772,809100,1047552,

%T 1335180,1678320,2083692,2558400,3109932,7308912,8500140,9831360,

%U 11313132,12956400,18970380,21376752,24005100,26868672,37008972,49780080

%N Numbers n such that the square part of n is one greater than the squarefree part of n.

%C The complementary sequence, squarefree part of n is one greater than the square part of n, is A069187.

%H Alois P. Heinz, <a href="/A189883/b189883.txt">Table of n, a(n) for n = 1..1000</a>

%H Antonio Roldán, <a href="http://hojaynumeros.blogspot.com">hojaynumeros.blogspot.com</a>

%F n such that A008833(n) - A007913(n) = 1.

%F a(n) = x^2 (x^2-1), where x = A067874(n). - _T. D. Noe_, Apr 29 2011

%e 1260 = 2^2*3^2*5*7, square part: 2^2*3^2 = 36, squarefree part: 5*7 = 35, and 36 = 35+1.

%p b:= proc() 1 end:

%p a:= proc(n) option remember; local i, k;

%p       if n>1 then a(n-1) fi;

%p       for k from b(n-1)+1 while 1<>mul(i[2], i=ifactors(k^2-1)[2])

%p       do od; b(n):=k; k^4-k^2

%p     end:

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Apr 29 2011

%t 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 (PARI) for(n=1,1e3,if(issquarefree(n^2-1),print1(n^4-n^2", "))) \\ _Charles R Greathouse IV_, Apr 29, 2011

%K nonn

%O 1,1

%A _Antonio Roldán_, Apr 29 2011