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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. 2
1, 2, 7, 7, 7, 43, 5, 16, 19, 87, 25, 31, 1061, 9, 43, 32815, 565, 63, 61, 16451, 7, 73, 1048655, 2131, 91, 97, 131173, 39, 107, 16777325, 4209, 127, 4294967427, 524425, 171, 149, 134217879, 4253, 163, 68719476903, 1048749, 187, 181, 536871103, 2241, 197, 549755814087 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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,...

Let the operator A_k map every k-th entry to its binary opposite (1->0, 0->1), for k=1,2,... .

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, ...

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.

The sequence lists numbers obtained by reading rows of T_inf(P) in binary and converting them to decimal.

LINKS

Peter J. C. Moses, Table of n, a(n) for n = 1..500

FORMULA

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.

T_inf(O) begins:

1

00

100

0010

00000

100000

0001000

00000001

.........

Now we have T_inf(P) = XNOR(T_0(P), T_inf(O)).

EXAMPLE

T_0(P) begins:

0

10

11

101

111

1011

1101

10001

........

T_inf(P) begins:

1

10

111

0111

00111

101011

0000101

00010000

000010011

0001010111

00000011001

000000011111

0010000100101

.............

MATHEMATICA

seq=Apply[BitXor, {Map[If[IntegerQ[Sqrt[#]], 1, 0]&, Range[Length[#]]], #}&[Flatten[Join[{{0}}, Map[IntegerDigits[Prime[#], 2, #+1]&, Range[50]]]]]];

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 *)

CROSSREFS

Cf. A000040, A000225, A247092.

Sequence in context: A248675 A068386 A021040 * A330479 A257960 A238734

Adjacent sequences:  A246550 A246551 A246552 * A246554 A246555 A246556

KEYWORD

nonn

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Nov 16 2014

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)