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

12, 240, 1260, 20592, 38220, 65280, 104652, 159600, 233772, 809100, 1047552, 1335180, 1678320, 2083692, 2558400, 3109932, 7308912, 8500140, 9831360, 11313132, 12956400, 18970380, 21376752, 24005100, 26868672, 37008972, 49780080

Offset: 1

Author

Antonio Roldán, Apr 29 2011

Keywords

• nonn

Comments

The complementary sequence, squarefree part of k is one greater than the square part of k, is A069187.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Antonio Roldán, hojaynumeros.blogspot.com

Programs

Maple

b:= proc() 1 end:
a:= proc(n) option remember; local i, k;
      if n>1 then a(n-1) fi;
      for k from b(n-1)+1 while 1<>mul(i[2], i=ifactors(k^2-1)[2])
      do od; b(n):=k; k^4-k^2
    end:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 29 2011

Mathematica

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 *)

PARI

for(n=1,1e3,if(issquarefree(n^2-1),print1(n^4-n^2", "))) \\ Charles R Greathouse IV, Apr 29 2011

Formula

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

a(n) = x^2 (x^2-1), where x = A067874(n). - T. D. Noe, Apr 29 2011