A165418 -id:A165418 - cp's OEIS frontend

A165417 a(0) = 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 0 <= k < n.

1, 1, 2, 1, 4, 4, 5, 2, 8, 8, 8, 9, 14, 14, 12, 4, 16, 16, 16, 17, 20, 16, 23, 20, 36, 36, 36, 37, 42, 42, 28, 8, 32, 32, 32, 33, 32, 36, 39, 36, 48, 48, 32, 42, 64, 60, 57, 44, 88, 88, 88, 89, 96, 88, 97, 96, 128, 128, 128, 130, 116, 116, 64, 16, 64, 64, 64, 65, 64, 68, 71, 68, 72

Offset: 0

Leroy Quet, Sep 17 2009

The distinct nonnegative values of the substrings of binary n is row n of table A119709.

a(2^n) = 2^n, for all n.

			9 in binary is 1001. The distinct nonnegative integers that occur as substrings in binary 9 are 0, 1, 2 (10 in binary), 4 (100 in binary), and 9 (1001 in binary). So a(9) = a(0)+a(1)+a(2)+a(4) = 1 + 1 + 2 + 4 = 8.

Cf. A119709, A165418.

Extended by Ray Chandler, Mar 13 2010

A175491 a(1)=1. a(n+1) = Sum_{k=1..n} a(b(k,n)), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 25, 35, 49, 64, 89, 122, 174, 235, 286, 334, 407, 473, 581, 690, 824, 976, 1206, 1449, 1811, 2183, 2718, 3306, 4173, 5070, 5659, 6071, 6769, 7279, 8137, 8716, 9765, 10587, 11907, 12940, 14631, 15649, 17600, 19231, 21729, 24004, 27228

Offset: 1

Leroy Quet, May 28 2010

			a(6)=11 because 5=(101)2 and
for k=1=(1)2 CS (1)2 and a(1)=1
for k=2=(10)2 CS (10)2=2 and a(2)=1
for k=3=(11)2 CS (1)2 and a(1)=1
for k=4=(100)2 CS (10)2=2 and a(2)=1
for k=5=(101)2 CS (101)2=5 and a(5)=7
and the sum of these 5 terms is 11.
(CS stands for "largest common substring is").

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first differences of the first 10000 terms
Rémy Sigrist, PARI program for A175491

Cf. A165418, A175466.

PARI
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More terms from Lars Blomberg, Feb 25 2016

A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 2, 4, 8, 11, 8, 8, 11, 8, 4, 8, 20, 34, 26, 34, 51, 40, 20, 20, 40, 51, 34, 26, 34, 20, 8, 16, 48, 96, 76, 118, 186, 152, 76, 96, 208, 281, 186, 152, 208, 124, 48, 48, 124, 208, 152, 186, 281, 208, 96, 76, 152, 186, 118, 76, 96, 48, 16, 32

Offset: 1

Rémy Sigrist, Feb 02 2023

This sequence is a variant of A165418.

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------------------
   1     1  N/A
   2     1  {1}
   3     1  {1}
   4     2  {1, 2}
   5     3  {1, 2, 3}
   6     3  {1, 2, 3}
   7     2  {1, 3}
   8     4  {1, 2, 4}
   9     8  {1, 2, 3, 4, 5}
  10    11  {1, 2, 3, 4, 5, 6}
  11     8  {1, 2, 3, 5, 7}
  12     8  {1, 2, 3, 4, 6}
  13    11  {1, 2, 3, 5, 6, 7}
  14     8  {1, 2, 3, 6, 7}
  15     4  {1, 3, 7}
  16     8  {1, 2, 4, 8}

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A165418, A301983.

{ for (n=1, #a=vector(64), print1 (a[n]=if (n==1, 1, s = [1]; b = binary(n); for (k=2, #b, s = setunion(s, apply(v -> 2*v+b[k], s))); sum(k=1, #s-1, a[s[k]]);)", ")) }

a(n) = Sum_{k = 1..A301977(n-1)} a(A301983(n, k)) for any n > 1.
a(2^k) = 2^(k-1) for any k > 0.
a(2^k-1) = 2^(k-2) for any k > 1.
a(n) >= A165418(n).

cp's OEIS Frontend

A165417 a(0) = 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 0 <= k < n.

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A175491 a(1)=1. a(n+1) = Sum_{k=1..n} a(b(k,n)), where b(k,n) is the largest positive integer that, when written in binary, occurs as a substring in both binary k and binary n.

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PARI

Extensions

A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

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