A078822 -id:A078822 - cp's OEIS frontend

A265236 Number of solutions to the equation A x B = C, where A, B and C are nonnegative numbers appearing as (contiguous) substrings of the binary representation of n.

1, 1, 8, 3, 13, 12, 18, 5, 19, 17, 18, 20, 31, 26, 28, 7, 26, 23, 23, 26, 31, 22, 32, 28, 47, 40, 38, 34, 49, 40, 38, 9, 34, 30, 29, 31, 31, 31, 38, 34, 47, 39, 28, 34, 53, 40, 46, 38, 66, 55, 54, 48, 59, 46, 46, 48, 75, 62, 58, 52, 67, 58, 48, 11, 43, 38

Offset: 0

Reinhard Zumkeller, Dec 06 2015

nonn

A, B and C are allowed to be zero, in contrast to A265008;
a(A000225(n)) = A265008(A000225(n));
a(A062289(n)) != A265008(A062289(n)).

			.  n | A007088 | A119709     |  a |
. ---+---------+-------------+----+-------------------------------------
.  2 |      10 | [0,1,2]     |  8 = #{(0,0,0), (0,1,0), (0,2,0), (1,0,0),
.    |         |             |        (2,0,0), (1,1,1), (1,2,2), (2,1,2)}
.  3 |      11 | [1,3]       |  3 = #{(1,1,1), (1,3,3), (3,1,3)}
.  4 |     100 | [0,1,2,4]   | 13 = #{(0,0,0), (0,1,0), (0,2,0), (0,4,0),
.    |         |             |         (1,0,0), (2,0,0), (4,0,0), (1,1,1),
.    |         |             |         (1,2,2), (2,1,2), (1,4,4), (2,2,4),
.    |         |             |         (4,1,4)}
.  5 |     101 | [0,1,2,5]   | 12 = #{(0,0,0), (0,1,0), (0,2,0), (0,5,0),
.    |         |             |         (1,0,0), (2,0,0), (5,0,0), (1,1,1),
.    |         |             |         (1,2,2), (2,1,2), (1,5,5), (5,1,5)}
.  6 |     110 | [0,1,2,3,6] | 18 = #{(0,0,0), (0,1,0), (0,2,0), (0,3,0),
.    |         |             |         (0,6,0), (1,0,0), (2,0,0), (3,0,0),
.    |         |             |         (6,0,0), (1,1,1), (1,2,2), (2,1,2),
.    |         |             |         (1,3,3), (3,1,3), (1,6,6), (2,3,6),
.    |         |             |         (3,2,6), (6,1,6)}
.  7 |     111 | [1,3,7]     |≈ 5 = #{(1,1,1), (1,3,3), (3,1,3), (1,7,7),
.    |         |             |         (7,1,7)} .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088, A119709, A265008, A265183 (decimal case), A000225, A062289, A043545, A078822.

a265236 n = length [() | let cs = a119709_row n, a <- cs, b <- cs, c <- cs,
                         a * b == c || c == 0 && a * b == 0]

For n > 0: a(n) = A265008(n) + A043545(n) * (2*A078822(n) - 1).

Suggested by N. J. A. Sloane.

A272851 Number of distinct nonzero Fibonacci numbers among the contiguous substrings of the binary digits of n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 4, 3, 5, 3, 2, 3, 3, 2, 3, 3, 4, 4, 4, 4, 3, 5, 5, 3, 5, 3, 2, 3, 3, 4, 4, 2, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 4, 4, 4, 3, 3, 5, 6, 5, 6, 4, 3, 5, 5, 3, 5, 3, 2, 3, 3, 3, 4, 4, 5, 4, 4, 3, 2, 3, 4, 3, 5, 3, 3, 4

Offset: 1

Marko Riedel, May 07 2016

			a(53) = 6 because 53=(110101)_2 which contains (1)_2 = 1, (10)_2 = 2, (11)_2 = 3, (101)_2 = 5, (1101)_2 = 13 and (10101)_2 = 21. The one digit only contributes once.

Marko Riedel, Maple program to compute sequence.

Cf. A000045, A078822, A272852, A272886.

s = Fibonacci@ Range@ 30; Table[Length@ Select[Union@ Flatten@ Function[k, Map[FromDigits[#, 2] & /@ Partition[k, #, 1] &, Range@ Length@ k]]@IntegerDigits[#, 2] &@ n, MemberQ[s, #] &], {n, 120}] (* Michael De Vlieger, May 08 2016 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)) ;
a(n) = {vb = binary(n); vf = []; for (i=1, #vb, for (j=1, #vb - i + 1, pvb = vector(j, k, vb[i+k-1]); f = subst(Pol(pvb), x, 2); if (f && isfib(f), vf = Set(concat(vf, f))););); #vf;} \\ Michel Marcus, May 08 2016

A301977 a(n) is the number of distinct positive numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 4, 6, 7, 6, 6, 7, 6, 4, 5, 8, 10, 9, 10, 12, 11, 8, 8, 11, 12, 10, 9, 10, 8, 5, 6, 10, 13, 12, 14, 17, 16, 12, 13, 18, 20, 17, 16, 18, 15, 10, 10, 15, 18, 16, 17, 20, 18, 13, 12, 16, 17, 14, 12, 13, 10, 6, 7, 12, 16, 15, 18, 22, 21, 16, 18

Offset: 1

Rémy Sigrist, Mar 30 2018

This sequence has similarities with A078822; there we consider consecutive digits, here not.

			The first terms, alongside the binary representations of n and of the numbers k whose binary digits appear in order in the binary representation of k, are:
  n  a(n)  bin(n)    bin(k)
  -- ----  ------    ------
   1    1       1    1
   2    2      10    1, 10
   3    2      11    1, 11
   4    3     100    1, 10, 100
   5    4     101    1, 10, 11, 101
   6    4     110    1, 10, 11, 110
   7    3     111    1, 11, 111
   8    4    1000    1, 10, 100, 1000
   9    6    1001    1, 10, 11, 100, 101, 1001
  10    7    1010    1, 10, 11, 100, 101, 110, 1010
  11    6    1011    1, 10, 11, 101, 111, 1011
  12    6    1100    1, 10, 11, 100, 110, 1100
  13    7    1101    1, 10, 11, 101, 110, 111, 1101
  14    6    1110    1, 10, 11, 110, 111, 1110
  15    4    1111    1, 11, 111, 1111
  16    5   10000    1, 10, 100, 1000, 10000
  17    8   10001    1, 10, 11, 100, 101, 1000, 1001, 10001
  18   10   10010    1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 10010
  19    9   10011    1, 10, 11, 100, 101, 111, 1001, 1011, 10011
  20   10   10100    1, 10, 11, 100, 101, 110, 1000, 1010, 1100, 10100

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A070939, A078822.

b:= proc(n) option remember; `if`(n=0, {0},
      map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))
    end:
a:= n-> nops(b(n))-1:
seq(a(n), n=1..72);  # Alois P. Heinz, Jan 26 2022

a(n) = my (b=binary(n), s=Set(1)); for (i=2, #b, s = setunion(s, Set(apply(v -> 2*v+b[i], s)))); return (#s)

a(2^n) = n + 1 for any n >= 0.
a(2^n - 1) = n for any n > 0.
a(2^n + k) = a(2^(n+1)-1 - k) for any n >= 0 and k=0..2^n-1.
a(n) >= A070939(n) for any n > 0.
a(n) = Sum_{k=1..n} (Stirling2(n+1,k) mod 2) (conjecture). - Ilya Gutkovskiy, Jul 04 2019

A078832 Smallest prime contained as binary substring in binary representation of n>1, a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Dec 08 2002

a(n)<=3 and for n>1: a(n)>=2 and a(n)=3 iff n=2^k-1, k>1.

a(n) = A225243(n,1). - Reinhard Zumkeller, Aug 14 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A078833, A078826, A078822, A007088, A020639.

Cf. A030308, A010051, A000225.

a078832 = head . a225243_row  -- Reinhard Zumkeller, Aug 14 2013

For n > 1: a(n) = A036987(n) + 2. Reinhard Zumkeller, Aug 14 2013

A078834 Greatest prime factor of n also contained as binary substring in binary representation of n; a(n)=1, if no such factor exists.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 1, 5, 11, 3, 13, 7, 3, 2, 17, 2, 19, 5, 1, 11, 23, 3, 1, 13, 3, 7, 29, 3, 31, 2, 1, 17, 1, 2, 37, 19, 3, 5, 41, 2, 43, 11, 5, 23, 47, 3, 1, 2, 3, 13, 53, 3, 11, 7, 3, 29, 59, 3, 61, 31, 7, 2, 1, 2, 67, 17, 1, 2, 71, 2, 73, 37, 5, 19, 1, 3, 79, 5, 1, 41, 83, 2, 5, 43

Offset: 1

Reinhard Zumkeller, Dec 08 2002

a(n) <= min{A078833(n), A006530(n)};
for n>1: a(n) = n iff n is prime.
a(A100484(n)) = A000040(n); a(A100368(n)) = A006530(A100368(n)). [Reinhard Zumkeller, Sep 19 2011]

			n=15=3*5 has two factors; only '11'=3 is contained in '1111'=15, therefore a(15)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A078833, A078822, A007088, A006530.

import Numeric (showIntAtBase)
import Data.List (find, isInfixOf)
import Data.Maybe (fromMaybe)
a078834 n = fromMaybe 1 $ find (\p -> showIntAtBase 2 ("01" !!) p ""
                          `isInfixOf` showIntAtBase 2 ("01" !!) n "") $
                 reverse $ a027748_row n
-- Reinhard Zumkeller, Sep 19 2011

a[n_] := Module[{bn, pp, sel}, bn = IntegerDigits[n, 2]; pp = FactorInteger[n][[All, 1]]; sel = Select[pp, MatchQ[bn, {_, Sequence @@ IntegerDigits[#, 2], _}] &]; If[sel == {}, 1, Max[sel]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 13 2013 *)

A165153 a(n) = the product of all distinct positive (nonzero) integers that, when written in binary, occur as substrings in the binary representation of n.

Original entry on oeis.org

1, 2, 3, 8, 10, 36, 21, 64, 72, 100, 330, 1728, 2340, 3528, 315, 1024, 1088, 1296, 4104, 8000, 2100, 43560, 53130, 331776, 388800, 608400, 694980, 4741632, 6650280, 1587600, 9765, 32768, 33792, 36992, 114240, 46656, 239760, 935712, 1120392

Offset: 1

Leroy Quet, Sep 05 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A119709, A078823.

Cf. A165416.

a165153 = product . a165416_row  -- Reinhard Zumkeller, Aug 14 2013

a(n) = product(A165416(n,k): k=1..A078822(n)-1). - Reinhard Zumkeller, Aug 14 2013

More terms from Sean A. Irvine, Nov 12 2009

A272886 Number of distinct Fibonacci numbers among the contiguous substrings of the binary digits of n.

Original entry on oeis.org

1, 3, 2, 3, 4, 4, 2, 4, 3, 4, 5, 4, 6, 4, 2, 4, 4, 3, 4, 4, 5, 5, 5, 5, 4, 6, 6, 4, 6, 4, 2, 4, 4, 5, 5, 3, 4, 4, 4, 5, 4, 5, 6, 5, 6, 5, 5, 5, 5, 4, 4, 6, 7, 6, 7, 5, 4, 6, 6, 4, 6, 4, 2, 4, 4, 4, 5, 5, 6, 5, 5, 4, 3, 4, 5, 4, 6, 4, 4, 5, 5, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 5, 6, 5, 5, 5

Offset: 1

Marko Riedel, May 08 2016

nonn

			a(53) = 7 because 53=(110101)_2 which contains (0)_2 = 0, (1)_2 = 1, (10)_2 = 2, (11)_2 = 3, (101)_2 = 5, (1101)_2 = 13 and (10101)_2 = 21. The one digit only contributes once as do two and zero.

Marko Riedel, Maple program to compute sequence.

Cf. A000045, A078822, A272851, A272852.

A078824 Number of distinct binary numbers contained as substrings in binary, circular formed representation of n and not longer than n.

Original entry on oeis.org

1, 1, 3, 2, 4, 6, 6, 3, 5, 8, 5, 10, 8, 10, 10, 4, 6, 10, 9, 13, 9, 12, 12, 15, 10, 13, 12, 15, 13, 15, 15, 5, 7, 12, 11, 16, 7, 16, 16, 19, 11, 16, 7, 19, 16, 12, 19, 21, 12, 16, 16, 19, 16, 19, 12, 21, 16, 19, 19, 21, 19, 21, 21, 6, 8, 14, 13, 19, 12, 19, 19, 23, 12, 17, 16, 24

Offset: 0

Reinhard Zumkeller, Dec 08 2002

For k>0: a(2^k-1)=k, a(2^k)=k+2;
for k>1: a(2^k+1)=a(3*2^(k-1))=k+2;
for k>0: a(2^k-1)=A078822(2^k-1), a(2^k)=A078822(2^k).

Cf. A078825, A078822, A007088.

A078828 Product of all primes contained as binary substrings in binary representation of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 10, 6, 63, 2, 2, 20, 330, 6, 390, 126, 1323, 2, 34, 4, 114, 20, 100, 660, 159390, 6, 6, 780, 12870, 126, 237510, 2646, 861273, 2, 2, 68, 102, 4, 740, 228, 2394, 20, 820, 200, 141900, 660, 42900, 318780, 157317930, 6, 102, 12, 342, 780, 206700

Offset: 0

Reinhard Zumkeller, Dec 08 2002

			n=7: product of the A078827(7)=3 primes as binary substrings in binary representation of 7 -> '111': a(7) = '11'*'11'*'111' = 3*3*7 = 63.

Cf. A078827, A078822, A007088.

cp's OEIS Frontend

A265236 Number of solutions to the equation A x B = C, where A, B and C are nonnegative numbers appearing as (contiguous) substrings of the binary representation of n.

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A272851 Number of distinct nonzero Fibonacci numbers among the contiguous substrings of the binary digits of n.

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A301977 a(n) is the number of distinct positive numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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A078832 Smallest prime contained as binary substring in binary representation of n>1, a(1)=1.

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A078834 Greatest prime factor of n also contained as binary substring in binary representation of n; a(n)=1, if no such factor exists.

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A165153 a(n) = the product of all distinct positive (nonzero) integers that, when written in binary, occur as substrings in the binary representation of n.

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A272886 Number of distinct Fibonacci numbers among the contiguous substrings of the binary digits of n.

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A078824 Number of distinct binary numbers contained as substrings in binary, circular formed representation of n and not longer than n.

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A078828 Product of all primes contained as binary substrings in binary representation of n.