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A144016

a(n) = the largest positive integer m such that the binary representations of all positive integers <= m are found within the binary representation of n.

44 in binary is 101100. In this string we find 1 (1 in decimal): (1)01100; 10 (2 in decimal): (10)1100; 11 (3 in decimal): 10(11)00; 100 (4 in decimal): 101(100); 101 (5 in decimal): (101)100; and 110 (6 in decimal): 10(110)0; but not 111 (7 in decimal). So a(44) = 6.