A242869 -id:A242869 - cp's OEIS frontend

A091065 Numbers having in binary representation no proper prefix that is also a suffix.

0, 1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 26, 28, 30, 32, 40, 44, 48, 50, 52, 56, 58, 60, 62, 64, 72, 80, 84, 88, 92, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 128, 144, 152, 160, 164, 168, 172, 176, 180, 184, 188, 192, 194, 196, 200, 202, 208

Offset: 1

Reinhard Zumkeller, Dec 17 2003

A091064(a(n)) = 0, complement of A091066.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A070939, A242869.

A091066 Numbers having in binary representation at least one proper prefix that is also a suffix.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 99, 101

Offset: 1

Reinhard Zumkeller, Dec 17 2003

A091064(a(n)) > 0, complement of A091065.

Includes all odd numbers > 1. - Robert Israel, Feb 05 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A070939, A082461, A242869.

f:= proc(n) local L,m;
   if n::odd then return true fi;
   L:= convert(n,base,2);
   for m from 2 to nops(L)-1 do
     if L[1..m] = L[-m..-1] then return true fi;
   od:
   false
end proc:
select(f, [$2..101]); # Robert Israel, Feb 05 2016

A091064 In binary representation: length of longest proper prefix of n, that is also a suffix.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 2, 1, 0, 1, 0, 3, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 2, 1, 3, 1, 2, 1, 0, 1, 4, 1, 0, 3, 2, 1, 0, 1, 0, 2, 0, 1, 3, 2, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 2, 1, 3, 1, 2, 1, 0, 4, 2, 1, 3, 1, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 4, 0, 3, 2, 1, 0, 1, 0, 2, 0, 1

Offset: 0

Reinhard Zumkeller, Dec 17 2003

a(n) <= A070939(n)-1 by definition;
a(2^k-1)=k-1; for k>0: a(2^k+1)=1; for k>2: a(2^k+2)=2;
a(A091065(n)) = 0, a(A091066(n)) > 0.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A242869.

A147755 a(n) = the largest integer that, when represented in binary, occurs at least twice as a substring in the binary representation of n.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 1, 2, 1, 1, 1, 3, 7, 0, 1, 2, 1, 2, 5, 2, 3, 1, 1, 2, 3, 3, 3, 7, 15, 0, 1, 2, 1, 4, 2, 2, 3, 2, 2, 10, 5, 2, 5, 3, 7, 1, 1, 2, 3, 2, 5, 6, 3, 3, 3, 3, 3, 7, 7, 15, 31, 0, 1, 2, 1, 4, 2, 2, 3, 4, 9, 2, 2, 4, 2, 3, 7, 2, 2, 2, 2, 10, 21, 5, 5, 2, 2, 5, 11, 3, 5, 7, 15, 1, 1, 2, 3, 4, 2, 6, 3, 2

Offset: 3

Leroy Quet, Nov 11 2008

The substrings (each equal to the binary representation of a(n)) may overlap in the binary representation of n.

			21 in binary is 10101. 101 (5 in decimal) occurs twice in 10101: (101)01 and 10(101). Since no larger integer (when represented in binary) occurs more than once within 10101, then a(21) = 5.

Alois P. Heinz, Table of n, a(n) for n = 3..8190

Cf. A147756, A242869.

Table[With[{d = IntegerDigits[n, 2]}, SelectFirst[Range[n, 0, -1], SequenceCount[d, IntegerDigits[#, 2], Overlaps -> True] >= 2 &]], {n, 3, 104}] (* Michael De Vlieger, Oct 30 2017 *)

Extended by Ray Chandler, Nov 14 2008

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A091065 Numbers having in binary representation no proper prefix that is also a suffix.

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A091066 Numbers having in binary representation at least one proper prefix that is also a suffix.

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A091064 In binary representation: length of longest proper prefix of n, that is also a suffix.

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A147755 a(n) = the largest integer that, when represented in binary, occurs at least twice as a substring in the binary representation of n.

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