A207360 - OEIS

%I #21 Jun 18 2024 10:01:19

%S 8,40,56,88,104,136,152,184,232,248,280,288,296,328,344,376,424,440,

%T 472,488,520,536,568,584,616,632,664,675,680,712,728,760,776,808,824,

%U 856,872,904,920,952,1016,1048,1064,1096,1112,1144,1160,1192,1208,1240,1256

%N Numbers n, not squarefree, satisfying A055231(n) = A055231(n + A055231(n)).

%C A055231(n) is the powerfree part of n.

%C This sequence is infinite because the numbers of the form n = 8p, where p is prime, are in the sequence : A055231(8p) = p and A055231(8p + p) = A055231(9p)  = p.

%C The numbers such that n and n+1 are a pair of consecutive powerful numbers (the again infinite A060355) are also in the sequence because A055231 (A060355(n)) = A055231(A060355 (n+1)) = 1.

%e 136 is in the sequence because A055231(136) = A055231(17*2^3) = 17, A055231(136 + 17) = A055231(153) = A055231(17*3^2) = 17.

%p isA013929 := proc(n)

%p         n>3 and not numtheory[issqrfree](n) ;

%p end proc:

%p isA207360 := proc(n)

%p        isA013929(n)  and (A055231(n)- A055231(n+ A055231(n))=0);

%p end proc:

%p for n from 1 to 5000 do

%p         if isA207360(n) then

%p             printf(`%d, `,n);

%p         end if;

%p end do: # (adapted from A140394).

%t rad[n_] := Times @@ FactorInteger[n][[All, 1]];

%t A055231[n_] := Denominator[n/rad[n]^2];

%t Select[Range[2000], !SquareFreeQ[#] && A055231[#] == A055231[# + A055231[#]]&] (* _Jean-François Alcover_, Jun 18 2024 *)

%o (PARI) isA013929(n)={

%o     (n>3) && !issquarefree(n)

%o }

%o isA207360(n)={

%o     isA013929(n) && ( A055231(n)-A055231(n+A055231(n)) ==0)

%o }

%o { for(n=1,1300, if(isA207360(n), print1(n" ") ) ; ) ;

%o } /* _R. J. Mathar_, Mar 12 2012 */

%Y Cf. A055231, A060355, A140394.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 17 2012