A091066 -id:A091066 - 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.

A242869 Largest integer m < n having a binary expansion that is a prefix and a suffix of the binary expansion of n; a(0)=0.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, May 24 2014

The prefix and the suffix are allowed to overlap.
a(n) <= A147755(n).
a(2^n) = 0.
a(2^n-1) = 2^(n-1)-1 for n>0.
a(n) = 0 iff n in { A091065 }.
a(n) > 1 iff n in { A091066 }.
A029837(a(n)+1) = A091064(n).

			a(91) = 11 because 91 = (1011)011_2 = 101(1011)_2 and 11 = 1011_2.
a(84) = 0 because 84 = 1010100_2, only the empty bitstring is a proper prefix and suffix.

Alois P. Heinz, Table of n, a(n) for n = 0..8192

Cf. A147755.

a:= proc(n) local m; m:=n;
      while m>1 do m:= iquo(m, 2);
        if m=irem(n, 2^(1+ilog2(m))) then return m fi
      od; 0
    end:
seq(a(n), n=0..100);

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.

A342378 List of numbers whose binary expansion is an instance of the Zimin pattern ABA.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 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

Peter Kagey, Mar 09 2021

This is A091066 with some terms of A020330 removed.

			The binary expansion of 182 is 10110110, which is an instance of the pattern ABA with A=10 and B=1101.

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, Sesquipower

Cf. A020330, A091066, A254128, A262312.

def ok(n):
  b = bin(n)[2:]
  for i in range(1, (len(b)+1)//2):
    if b[:i] == b[-i:]: return True
  return False
def aupto(lim): return [m for m in range(lim+1) if ok(m)]
print(aupto(101)) # Michael S. Branicky, Mar 09 2021

cp's OEIS Frontend

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

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A242869 Largest integer m < n having a binary expansion that is a prefix and a suffix of the binary expansion of n; a(0)=0.

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Maple

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

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A342378 List of numbers whose binary expansion is an instance of the Zimin pattern ABA.

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Python