A247092 - OEIS

%I #26 Dec 11 2015 12:11:15

%S 1,0,4,2,0,32,8,1,0,64,4,0,1024,32,0,32768,512,4,0,16384,64,0,1048576,

%T 2048,2,0,131072,64,0,16777216,4096,0,4294967296,524288,32,0,

%U 134217728,4096,0,68719476736,1048576,8,0,536870912,2048,0,549755813888,1048576,1,0

%N Limiting sequence obtained by taking the sequence of Mersenne numbers 2^n-1, n=1,2,...(A000225) and applying an infinite process which is described in the comments.

%C Write the Mersenne numbers 2^n-1, n=1,2,..., in binary in form of triangle T_0(M) consisting of all 1's:

%C 1

%C 11

%C 111

%C 1111

%C 11111

%C ......

%C Let the operator A_k map every k-th entry to its binary opposite (1->0, 0->1), for k=2,3,... . Put T_inf(M) = ...*A_4*A_3*A_2(T_(0)M), with successive applications of the operators A_2, A_3, A_4, ...

%C Note that the (0,1)-triangle T_inf(M) is well-defined, since the operator T_n does not affect entries in the first floor((sqrt(8*n-7) - 1)/2) rows.

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

%C The n-th entry t_n of T_inf(M) equals 1, if n is perfect square, and 0 otherwise (A010052, for n>=1).

%C Indeed, in order to get an entry t_n of T_inf(M), we should use all considered operators A_d, d|n, d>1. The number of these operators is the diminished on 1 the number of divisors of n which is even iff n is a perfect square. Thus only in this case we obtain that entry in the n-th position is flipped, beginning with 1, an even number of times, such that t_n=1, while, if n is nonsquare, t_n=0.

%C Note that, since (n+1)^2 - n^2 > n, then in every row of T_inf(M) there exists at most one 1. So every term is either 0 or a power of 2.

%H Charles R Greathouse IV, <a href="/A247092/b247092.txt">Table of n, a(n) for n = 1..3453</a> (all terms less than 10^1000)

%F If there is an integer x in [sqrt((n-1)*n/2 +1), sqrt(n*(n+1)/2)] then it is unique and a(n) = 2^(n(n+1)/2-x^2); otherwise, a(n)=0.

%F Thus there are n/sqrt(2) + O(1) positive terms among the first n.

%e T_inf(M) begins

%e 1

%e 00

%e 100

%e 0010

%e 00000

%e 100000

%e 0001000

%e 00000001

%e .........

%e Let n=4. Then the interval in the formula is [sqrt(7), sqrt(10)], so x=3 and a(4) = 2^(10-9) = 2.

%o (PARI) a(n)=my(mx=n*(n+1)/2,s=sqrtint(mx)^2); if(s<=mx-n, 0, 2^(mx-s)) \\ _Charles R Greathouse IV_, Nov 19 2014

%Y Cf. A000225, A010052, A246553.

%K nonn,base,easy

%O 1,3

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Nov 18 2014

%E a(25)-a(50) from _Charles R Greathouse IV_, Nov 19 2014