A101933 Concatenate the first n positive integers written in binary (with each integer written so the most significant 1 is on the left and the concatenated string is from left to right, A030302). a(n) is the number of times n written in binary appears disjointedly in the concatenated string.

1, 1, 2, 1, 2, 3, 3, 1, 2, 1, 4, 3, 4, 5, 4, 1, 2, 2, 3, 2, 2, 1, 6, 3, 4, 4, 6, 6, 5, 7, 5, 1, 2, 2, 3, 1, 4, 3, 4, 2, 4, 2, 5, 2, 3, 4, 8, 3, 4, 5, 5, 4, 6, 4, 8, 6, 6, 6, 7, 9, 6, 9, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 5, 2, 6, 4, 5, 2, 4, 3, 6, 3, 3, 3, 7, 2, 5, 3, 5, 3, 7, 5, 10, 3, 4, 5, 5, 4, 7, 5, 7, 4, 6

Offset: 1

Author

Diana L. Mecum and Robert G. Wilson v, Aug 31 2008

Keywords

• base
• easy
• nonn

Comments

First occurrence of n: 1, 3, 6, 11, 14, 23, 30, 47, 60, 95, 126, 124, 223, 248, 252, 447, 863, 504, 895, 1008, 1020, 1016, 1919, 2044, 2032, 2040, 3839, 7159, 7871, 4080, 7679, 15855, 8160, 8184, 8176, 15839, ??, 16376, 16352, 16368, ..., to a limit of 25000.

Examples

			a(21) = 2 because 21_10 = 10101_2 and that appears in
1101110010111011110001001101010111100110111101111100001000110010100111010010101_2
three times but only two times which do not overlap.
		

Links

• Diana Mecum & Robert G. Wilson v, Table of n, a(n) for n = 1..10000.

Crossrefs

Cf. A007088, A030190, A030302, A058935, A047778. Differs from A117127 at positions in A141439.

Programs

Mathematica

a[n_] := StringCount[ ToString@ FromDigits@ Flatten@ IntegerDigits[ Range@n, 2], ToString@ FromDigits@ IntegerDigits[n, 2], Overlaps -> False]; Array[a, 105]