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Odd numbers k such that A163511(k) is a square.
1

%I #9 Oct 01 2023 11:17:04

%S 5,11,17,23,35,41,47,65,71,83,89,95,131,137,143,161,167,179,185,191,

%T 257,263,275,281,287,323,329,335,353,359,371,377,383,515,521,527,545,

%U 551,563,569,575,641,647,659,665,671,707,713,719,737,743,755,761,767,1025,1031,1043,1049,1055,1091,1097,1103,1121

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

%C The sequence is defined inductively as follows:

%C (a) it contains all numbers of the form 2^(2e) + 1, e >= 1,

%C and

%C (b) for any term a(n) and any e >= 0, (2^(2e+1) * a(n)) + 1 is also included as a term. There are no other kind of terms.

%C Contains no squares. See A365808 for a proof.

%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 isA365809(n) = (n%2 && issquare(A163511(n)));

%o (PARI)

%o A209229(n) = (n && !bitand(n,n-1));

%o isA365809(n) = if(n<=2,0,my(v=valuation(n-1,2)); if(A209229(n-1),!(v%2),(v%2)&&isA365809(n>>v)));

%Y Odd terms of A365808.

%Y Cf. A016754, A052539 (subsequence without its initial 2), A163511.

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 01 2023