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A122953
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a(n) = number of distinct positive integers represented in binary which are substrings of binary expansion of n.
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13
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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
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1,2
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COMMENTS
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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 n-th row in triangle A165416. - Reinhard Zumkeller, Jul 17 2015
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LINKS
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Jeremy Gardiner, Table of n, a(n) for n = 1..2000
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EXAMPLE
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Binary 1 = 1, binary 2 = 10, binary 4 = 100 and binary 9 = 1001 are all substrings of binary 9 = 1001. So a(9) = 4.
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MAPLE
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a:= n-> (s-> nops({seq(seq(parse(s[i..j]), i=1..j),
j=1..length(s))} minus {0}))(""||(convert(n, binary))):
seq(a(n), n=1..100); # Alois P. Heinz, Jan 20 2021
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MATHEMATICA
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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]
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PROG
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(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
(Python)
def a(n):
b = bin(n)[2:]
m = len(b)
return len(set(int(b[i:j]) for i in range(m) for j in range(i+1, m+1))-{0})
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jan 20 2021
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CROSSREFS
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Cf. A078822, A292924.
Cf. A057427, A213629, A165416.
Sequence in context: A117119 A208280 A139141 * A259847 A259103 A334200
Adjacent sequences: A122950 A122951 A122952 * A122954 A122955 A122956
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KEYWORD
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nonn,base
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AUTHOR
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Leroy Quet, Oct 25 2006
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EXTENSIONS
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More terms from Robert G. Wilson v, Nov 01 2006
Keyword base added by Rémy Sigrist, Mar 08 2018
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STATUS
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approved
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