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A365808 Numbers k such that A163511(k) is a square.

%I A365808 #21 Feb 12 2025 19:05:29
%S A365808 0,2,5,8,11,17,20,23,32,35,41,44,47,65,68,71,80,83,89,92,95,128,131,
%T A365808 137,140,143,161,164,167,176,179,185,188,191,257,260,263,272,275,281,
%U A365808 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 A365808 Numbers k such that A163511(k) is a square.
%C A365808 The sequence is defined inductively as:
%C A365808  (a) it contains 0 and 2,
%C A365808    and
%C A365808  (b) for any nonzero term a(n), (2*a(n)) + 1 and 4*a(n) are also included as terms.
%C A365808 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 A365808 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%o A365808 (PARI)
%o A365808 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 A365808 isA365808v2(n) = issquare(A163511(n));
%o A365808 (PARI) isA365808(n) = if(n<=2, !(n%2), if(n%2, isA365808((n-1)/2), if(n%4, 0, isA365808(n/4))));
%o A365808 (Python)
%o A365808 from itertools import count, islice
%o A365808 def A365808_gen(): # generator of terms
%o A365808     return map(lambda n:(3*(n+1)>>2)-1,filter(lambda n:n==1 or (n&3==3 and not '00' in bin(n)),count(1)))
%o A365808 A365808_list = list(islice(A365808_gen(),20)) # _Chai Wah Wu_, Feb 12 2025
%Y A365808 Cf. A000290, A010052, A032766, A163511, A365807 (characteristic function).
%Y A365808 Positions of even terms in A365805.
%Y A365808 Sequence A243071(n^2), n >= 1, sorted into ascending order.
%Y A365808 Subsequences: A004171, A055010, A365809 (odd terms).
%Y A365808 Subsequence of A016789 (after the initial 0).
%Y A365808 Cf. also A277335, A365801, A365802, A365806.
%K A365808 nonn
%O A365808 1,2
%A A365808 _Antti Karttunen_, Oct 01 2023