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A165418
a(1) = 1. For n >= 2, a(n) = sum a(k), where k is over the distinct values of the substrings in binary n, and where 1 <= k < n.
4
1, 1, 1, 2, 2, 3, 2, 4, 4, 4, 5, 8, 8, 8, 4, 8, 8, 8, 9, 10, 8, 13, 12, 20, 20, 20, 21, 26, 26, 20, 8, 16, 16, 16, 17, 16, 18, 21, 20, 24, 24, 16, 22, 36, 34, 35, 28, 48, 48, 48, 49, 52, 48, 55, 56, 76, 76, 76, 78, 76, 76, 48, 16, 32, 32, 32, 33, 32, 34, 37, 36, 36, 32, 40, 42, 50, 52
OFFSET
1,4
COMMENTS
The distinct positive values of the substrings of binary n is row n of table A165416.
a(2^n) = 2^(n-1), for all n >=1.
LINKS
EXAMPLE
9 in binary is 1001. The distinct positive integers that occur as substrings in binary 9 are 1, 2 (10 in binary), 4 (100 in binary), and 9 (1001 in binary). So a(9) = a(1)+a(2)+a(4) = 1 + 1 + 2 = 4.
MATHEMATICA
Fold[Function[{a, n}, Append[a, With[{w = IntegerDigits[n, 2]}, Total@ Part[a, Select[Union@ Map[FromDigits[#, 2] &, Apply[Join, Array[Partition[w, #, 1] &, Length@ w]]], Nor[# == 0, # == n] &]]]]], {1}, Range[2, 77]] (* Michael De Vlieger, Dec 31 2017 *)
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Leroy Quet, Sep 17 2009
EXTENSIONS
More terms from Sean A. Irvine, Nov 19 2009
STATUS
approved