A365808 - OEIS

%I #18 Oct 01 2023 11:17:09

%S 0,2,5,8,11,17,20,23,32,35,41,44,47,65,68,71,80,83,89,92,95,128,131,

%T 137,140,143,161,164,167,176,179,185,188,191,257,260,263,272,275,281,

%U 284,287,320,323,329,332,335,353,356,359,368,371,377,380,383,512,515,521,524,527,545,548,551,560,563,569,572,575

%N Numbers k such that A163511(k) is a square.

%C The sequence is defined inductively as:

%C  (a) it contains 0 and 2,

%C    and

%C  (b) for any nonzero term a(n), (2*a(n)) + 1 and 4*a(n) are also included as terms.

%C Because the inductive definition guarantees that all terms after 0 are of the form 3k+2 (A016789), and because for any n >= 0, n^2 == 0 or 1 (mod 3), (i.e., squares are in A032766), it follows that there are no squares in this sequence after the initial 0.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%o (PARI)

%o A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));

%o isA365808v2(n) = issquare(A163511(n));

%o (PARI) isA365808(n) = if(n<=2, !(n%2), if(n%2, isA365808((n-1)/2), if(n%4, 0, isA365808(n/4))));

%Y Cf. A000290, A010052, A032766, A163511, A365807 (characteristic function).

%Y Positions of even terms in A365805.

%Y Sequence A243071(n^2), n >= 1, sorted into ascending order.

%Y Subsequences: A004171, A055010, A365809 (odd terms).

%Y Subsequence of A016789 (after the initial 0).

%Y Cf. also A277335, A365801, A365802, A365806.

%K nonn

%O 1,2

%A _Antti Karttunen_, Oct 01 2023