cp's OEIS Frontend

A168157 Number of 0's in the matrix whose lines are the binary expansion of the first n primes.

%I A168157 #3 Jul 14 2012 11:32:32
%S A168157 1,1,4,4,9,10,19,21,22,23,23,37,40,42,43,45,46,47,69,72,76,78,81,84,
%T A168157 88,91,93,95,97,100,100,136,141,145,149,152,155,159,162,165,168,171,
%U A168157 172,177,181,184,187,188,191,194,197,198,201,202,263,268,273,277,282,287
%N A168157 Number of 0's in the matrix whose lines are the binary expansion of the first n primes.
%C A168157 The matrix is to be taken of minimal size, i.e., have n lines and the number of columns needed to write the n-th prime in the last line, A035100(n). Otherwise said, there is no zero column except for n=1 (prime(1) = 2 = 10[2] in binary).
%C A168157 The number of zeros in the last line of the matrix is given by A035103(n).
%C A168157 One has a(n)=a(n-1) iff n = A059305(k) for some k, i.e. prime(n) is a Mersenne prime A000668(k) = A000225(A000043(k)).
%C A168157 If prime(n)=2^2^k+1 is a Fermat prime (A019434), n>2, then one has a(n)=a(n-1)+n-1+2^k-1.
%C A168157 More generally, the "big jumps" a(n+1) > a(n)+n happen whenever a column is added, i.e. when prime(n) = A014234(k) <=> prime(n+1) = A104080(k) for some k,n>1.
%F A168157 a(n)=n*A035100(n)-A095375(n).
%e A168157 a(4)=4 is the number of zeros in the matrix [010] /* = 2 in binary */ [011] /* = 3 in binary */ [101] /* = 5 in binary */ [111] /* = 7 in binary */
%o A168157 (PARI) A168157(n)=n*#binary(prime(n))-sum(i=1,n,norml2(binary(prime(i))))
%K A168157 base,nonn
%O A168157 1,3
%A A168157 _M. F. Hasler_, Nov 21 2009