

A122953


a(n) = number of distinct positive integers represented in binary which are substrings of binary expansion of n.


13



1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 5, 6, 6, 6, 4, 5, 5, 5, 6, 6, 5, 7, 7, 8, 8, 8, 8, 9, 9, 8, 5, 6, 6, 6, 7, 6, 7, 8, 8, 8, 8, 6, 8, 10, 9, 10, 9, 10, 10, 10, 10, 11, 10, 10, 11, 12, 12, 12, 12, 12, 12, 10, 6, 7, 7, 7, 8, 7, 8, 9, 9, 8, 7, 9, 10, 10, 11, 11, 10, 10, 10, 10, 11, 9, 7, 11, 11, 13, 13, 12
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OFFSET

1,2


COMMENTS

a(n) = A078822(n) if n is of the form 2^k  1. Otherwise, a(n) = A078822(n)  1.
First occurrence of k: 1, 2, 4, 6, 11, 12, 22, 24, 28, 44, 52, 56, 88, 92, 112, 116, 186, 184, 220, 232, 244, 368, 376, 440, 472, ... (See A292924 for the corresponding sequence.  Rémy Sigrist, Mar 09 2018)
Last occurrence of k: 2^k  1.
a(n) = sum (A057427(A213629(n,k): k = 1 .. n).  Reinhard Zumkeller, Jun 17 2012
Length of nth row in triangle A165416.  Reinhard Zumkeller, Jul 17 2015


LINKS

Jeremy Gardiner, Table of n, a(n) for n = 1..2000


EXAMPLE

Binary 1 = 1, binary 2 = 10, binary 4 = 100 and binary 9 = 1001 are all substrings of binary 9 = 1001. So a(9) = 4.


MATHEMATICA

f[n_] := Length@ Select[ Union[ FromDigits /@ Flatten[ Table[ Partition[ IntegerDigits[n, 2], i, 1], {i, Floor[ Log[2, n] + 1]}], 1]], # > 0 &]; Array[f, 90]


PROG

(Haskell)
a122953 = length . a165416_row
 Reinhard Zumkeller, Jul 17 2015, Jan 22 2012
(PARI) a(n) = my (v=0, s=0, x=Set()); while (n, my (r=n); while (r, if (r < 100 000, if (bittest(s, r), break, s+=2^r), if (setsearch(x, r), break, x=setunion(x, Set(r)))); v++; r \= 2); n = 2^(#binary(n)1)); v \\ Rémy Sigrist, Mar 08 2018


CROSSREFS

Cf. A078822, A292924.
Cf. A057427, A213629, A165416.
Sequence in context: A117119 A208280 A139141 * A259847 A259103 A334200
Adjacent sequences: A122950 A122951 A122952 * A122954 A122955 A122956


KEYWORD

nonn,base


AUTHOR

Leroy Quet, Oct 25 2006


EXTENSIONS

More terms from Robert G. Wilson v, Nov 01 2006
Keyword base added by Rémy Sigrist, Mar 08 2018


STATUS

approved



