cp's OEIS Frontend

A357409 a(n) is the maximum number of positive numbers in a set of n consecutive positive or negative odd numbers such that the number of pairs that add to a power of 2 is maximal.

%I A357409 #39 Oct 27 2022 05:33:49
%S A357409 1,2,3,3,4,5,5,6,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,17,
%T A357409 18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,27,27,28,28,29,
%U A357409 29,30,30,31,31,32,33,33,34,34,35,35,36,36,37,37,38,38,39
%N A357409 a(n) is the maximum number of positive numbers in a set of n consecutive positive or negative odd numbers such that the number of pairs that add to a power of 2 is maximal.
%C A357409 For this sequence we check the sums for all possible pairs between two distinct members of a set of consecutive odd numbers, we count as valid results of the form 2^m, valid sums are obviously >= 1.
%C A357409 The related sequence A357574 counts the pairs that add to a power of 2.
%C A357409 For sequences of consecutive odd numbers the maximum number of pairs that add to a power of 2 will never be obtained if the sequence starts with numbers greater than 1. Obviously for sequences of all negative numbers, we will not obtain any valid pair.
%C A357409 It appears likely that an optimal solution for A352178 will contain only odd numbers as a mix of odd and even numbers would lead to two unconnected graphs. Let us now assume A352178(k) has an optimal solution in odd numbers only, then there exists a least m such that a(k+m) uses a superset of the same solution pairs used in A352178(k). It would be interesting to know if m has a bound in relation to k.
%H A357409 Thomas Scheuerle, <a href="/A357409/a357409.png">The first differences of the first 501 elements of this sequence.</a>
%F A357409 Conjecture: a(n) = A274089(n) + 2^t - 1, where t is the number of solutions k > 0 to the inequality n >= 2^(2*(k+1) + 1) - 2^(k+1) - 1. For n < 27, (2^t - 1) is 0, it is 1 for 27 <= n < 119, and it is 3 for 119 <= n < 495.
%e A357409 a(5) = 4 because {1, 3, 5, 7, 9} has four valid pairs: 1+3, 1+7, 3+5, 7+9. But {-1, 1, 3, 5, 7} has only four positive numbers and five valid pairs: 1+3, 1+7, 3+5, 3-1, 5-1, and there is no other set of consecutive numbers with 4 sums being a power of 2.
%o A357409 (MATLAB)
%o A357409 function a = A357409( max_n )
%o A357409     a(1) = 1; q = [];
%o A357409     for n = 1:max_n
%o A357409         c = 0;
%o A357409         for k = 0:n
%o A357409             s = (2*([0:n]-k))+1;
%o A357409             r = countpowtwo(s);
%o A357409             if c < r
%o A357409                 c = r;
%o A357409                 q = s;
%o A357409             end
%o A357409         end
%o A357409         a(n+1) = length(find(q > 0));
%o A357409     end
%o A357409 end
%o A357409 function c = countpowtwo(s)
%o A357409     M = repmat(s,[length(s),1]);
%o A357409     M = M+M';
%o A357409     M(M<=0) = 7;
%o A357409     M = bitand(M,M-1);
%o A357409     M = M + eye(size(M));
%o A357409     c = length(find(M == 0))/2;
%o A357409 end
%Y A357409 Cf. A274089, A347301, A352178, A357574.
%K A357409 nonn
%O A357409 1,2
%A A357409 _Thomas Scheuerle_, Sep 26 2022