A140394 -id:A140394 - cp's OEIS frontend

A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

9, 45, 63, 99, 117, 153, 171, 207, 261, 279, 289, 315, 333, 369, 387, 423, 477, 495, 531, 549, 585, 603, 639, 657, 676, 693, 711, 747, 765, 801, 819, 855, 873, 909, 927, 963, 981, 1017, 1035, 1071, 1143, 1179, 1197, 1233, 1251, 1287, 1305, 1341, 1359, 1395

Offset: 1

Michel Lagneau, Feb 17 2012

nonn

A055231(n) is the powerfree part of n. This sequence is infinite because all numbers of the form n = 9p, where p is a prime > 3, are in the sequence : A055231(9p) = p and A055231(9p - p) = A055231(8p) = p. The positive numbers of A055792 are also in the sequence because A055792(n) are squares and A055792(n)-1 are also squares.

			63 is in the sequence because A055231(63) = A055231(7*3^2) = 7, A055231(63 - 7) = A055231(56) = A055231(7*2^3) = 7.

Cf. A055231, A140394.

isA013929 := proc(n)
    n>3 and not numtheory[issqrfree](n) ;
end proc:
isA207359 := proc(n)
    isA013929(n) and (A055231(n)- A055231(n- A055231(n))=0);
end proc:
for n from 1 to 5000 do
      if isA207359(n) then
          printf(`%d, `,n);
      end if;
end do: #  adapted from A140394.

A207360 Numbers n, not squarefree, satisfying A055231(n) = A055231(n + A055231(n)).

Original entry on oeis.org

8, 40, 56, 88, 104, 136, 152, 184, 232, 248, 280, 288, 296, 328, 344, 376, 424, 440, 472, 488, 520, 536, 568, 584, 616, 632, 664, 675, 680, 712, 728, 760, 776, 808, 824, 856, 872, 904, 920, 952, 1016, 1048, 1064, 1096, 1112, 1144, 1160, 1192, 1208, 1240, 1256

Offset: 1

Michel Lagneau, Feb 17 2012

nonn

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

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

Cf. A055231, A060355, A140394.

isA013929 := proc(n)
        n>3 and not numtheory[issqrfree](n) ;
end proc:
isA207360 := proc(n)
       isA013929(n)  and (A055231(n)- A055231(n+ A055231(n))=0);
end proc:
for n from 1 to 5000 do
        if isA207360(n) then
            printf(`%d, `,n);
        end if;
end do: # (adapted from A140394).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A055231[n_] := Denominator[n/rad[n]^2];
Select[Range[2000], !SquareFreeQ[#] && A055231[#] == A055231[# + A055231[#]]&] (* Jean-François Alcover, Jun 18 2024 *)

isA013929(n)={
    (n>3) && !issquarefree(n)
}
isA207360(n)={
    isA013929(n) && ( A055231(n)-A055231(n+A055231(n)) ==0)
}
{ for(n=1,1300, if(isA207360(n), print1(n" ") ) ; ) ;
} /* R. J. Mathar, Mar 12 2012 */

cp's OEIS Frontend

A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

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