cp's OEIS Frontend

A246553 Limiting sequence obtained by taking the sequence 0, 2, 3, 5, 7, 11, 13, ... and applying an infinite process which is described in the comments.

%I A246553 #75 Feb 15 2016 14:53:15
%S A246553 1,2,7,7,7,43,5,16,19,87,25,31,1061,9,43,32815,565,63,61,16451,7,73,
%T A246553 1048655,2131,91,97,131173,39,107,16777325,4209,127,4294967427,524425,
%U A246553 171,149,134217879,4253,163,68719476903,1048749,187,181,536871103,2241,197,549755814087
%N A246553 Limiting sequence obtained by taking the sequence 0, 2, 3, 5, 7, 11, 13, ... and applying an infinite process which is described in the comments.
%C A246553 Consider the (0,1)-triangle T_0(P) with entries numbered 1,2,3,4,..., the rows of which, read in binary and converted to decimal, give the sequence 0 followed by consecutive primes, 0,2,3,5,7,...
%C A246553 Let the operator A_k map every k-th entry to its binary opposite (1->0, 0->1), for k=1,2,... .
%C A246553 Put T_inf(P) = ...*A_3*A_2*A_1(T_(0)P), with successive applications of the operators A_1, A_2, A_3, ...
%C A246553 Note that the (0,1)-triangle T_inf(P) is well-defined, since the operator T_n does not affect entries in the first floor((sqrt(8*n-7) - 1)/2) rows.
%C A246553 The sequence lists numbers obtained by reading rows of T_inf(P) in binary and converting them to decimal.
%H A246553 Peter J. C. Moses, <a href="/A246553/b246553.txt">Table of n, a(n) for n = 1..500</a>
%F A246553 If we take the initial triangle T_0(O) to consist of all 0's, then in T_inf(O) the 1's are only on positions of squares of all positive numbers, i.e., 1,4,9,16,... . Indeed, in order to get an entry in the n-th position of T_inf(O), we should use all considered operators A_d, d|n. The number of these operators is the number of divisors of n which is odd iff n is a perfect square. Thus only in this case, we obtain that entry in the n-th position is flipped, beginning with 0, an odd number of times, such that in the n-th position of T_inf(O) we have 1, while, if n is nonsquare, in the n-th position we have 0.
%F A246553 T_inf(O) begins:
%F A246553 1
%F A246553 00
%F A246553 100
%F A246553 0010
%F A246553 00000
%F A246553 100000
%F A246553 0001000
%F A246553 00000001
%F A246553 .........
%F A246553 Now we have T_inf(P) = XNOR(T_0(P), T_inf(O)).
%e A246553 T_0(P) begins:
%e A246553 0
%e A246553 10
%e A246553 11
%e A246553 101
%e A246553 111
%e A246553 1011
%e A246553 1101
%e A246553 10001
%e A246553 ........
%e A246553 T_inf(P) begins:
%e A246553 1
%e A246553 10
%e A246553 111
%e A246553 0111
%e A246553 00111
%e A246553 101011
%e A246553 0000101
%e A246553 00010000
%e A246553 000010011
%e A246553 0001010111
%e A246553 00000011001
%e A246553 000000011111
%e A246553 0010000100101
%e A246553 .............
%t A246553 seq=Apply[BitXor,{Map[If[IntegerQ[Sqrt[#]],1,0]&,Range[Length[#]]],#}&[Flatten[Join[{{0}},Map[IntegerDigits[Prime[#],2,#+1]&,Range[50]]]]]];
%t A246553 Map[FromDigits[#,2]&,MapThread[seq[[#1;;#2]]&,({Join[{0},Most[#1]]+1,#1}&)[#/2(#+1)&[Range[NestWhile[#+1&,1,((1+#1) (2+#1)<=2Length[seq])&]]]]]] (* _Peter J. C. Moses_, Nov 18 2014 *)
%Y A246553 Cf. A000040, A000225, A247092.
%K A246553 nonn
%O A246553 1,2
%A A246553 _Vladimir Shevelev_ and _Peter J. C. Moses_, Nov 16 2014