A062289 -id:A062289 - cp's OEIS frontend

A138890 Non-Padovan numbers.

6, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Omar E. Pol, Apr 05 2008

Natural numbers that are not in the Padovan sequence A000931.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A000931, A001690, A062289, A092460, A134816, A138836, A138888, A138891.

Complement[Range[0, Max[#]], #] &@ Union@ LinearRecurrence[{0, 1, 1}, {1, 0, 0}, 23] (* Michael De Vlieger, Sep 17 2024 *)

def A138890(n):
    def f(x):
        if x<=1: return n+1
        a, b, c, d = 1, 1, 1, 0
        while c<=x:
            a, b, c = b, c, a+b
            d += 1
        return n+d-1
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

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.

Original entry on oeis.org

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.

A138891 Non-Motzkin numbers.

Original entry on oeis.org

0, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Omar E. Pol, Apr 05 2008

Nonnegative integers that are not in A001006.

Cf. A001006, A001690, A062289, A092460, A138836, A138888, A138890.

A138887 Numbers that are not Sophie Germain primes.

Original entry on oeis.org

0, 1, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Omar E. Pol, Apr 05 2008

Nonnegative integers that are not in A005384.

A156660(a(n)) = 0; A053176 is a subsequence. [From Reinhard Zumkeller, Feb 18 2009]

Cf. A001690, A005384, A053176, A062289, A138888, A138889, A138890, A138891.

A257130 Where the difference A055938(n) - A005187(n) obtains record values; positions of records in A257126.

Original entry on oeis.org

1, 2, 5, 11, 23, 27, 55, 111, 121, 245, 247, 495, 503, 1007, 2015, 2037, 4077, 8157, 8175, 8179, 16363, 16367, 32735, 65471, 65517, 65519, 131039, 131055, 262111

Offset: 1

Antti Karttunen, Apr 16 2015

nonn

The corresponding record values of A257126 are 1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, ..., (possibly A062289).

Cf. A005187, A055938, A062289, A257126.

A276194 Odd numbers whose binary representation contains an even number of 1's and at least one 0.

Original entry on oeis.org

5, 9, 17, 23, 27, 29, 33, 39, 43, 45, 51, 53, 57, 65, 71, 75, 77, 83, 85, 89, 95, 99, 101, 105, 111, 113, 119, 123, 125, 129, 135, 139, 141, 147, 149, 153, 159, 163, 165, 169, 175, 177, 183, 187, 189, 195, 197, 201, 207, 209, 215, 219, 221, 225, 231, 235, 237

Offset: 1

Lei Zhou, Oct 20 2016

			Binary expansions of odd integers in decimal and binary forms are as follows:
   1 ->     1, no;
   3 ->    11, no;
   5 ->   101, yes, so a(1)=5;
   7 ->   111, no;
   9 ->  1001, yes so a(2)=9;
  11 ->  1011, no;
  13 ->  1101, no;
  15 ->  1111, no;
  17 -> 10001, yes so a(3)=17.

Cf. A005408.

Intersection of A129771 and A062289.

BNDigits[m_Integer] :=
  Module[{n = m, d, t = {}},
   While[n > 0, d = Mod[n, 2]; PrependTo[t, d]; n = (n - d)/2]; t];
c = 1;
Table[While[c = c + 2; d = BNDigits[c]; ld = Length[d];
  c1 = Total[d]; ! (EvenQ[c1] && (c1 < ld))]; c, {n, 1, 57}]

isok(n) = my(b=binary(n)); (n % 2) && (vecmin(b)==0) && !(vecsum(b) % 2); \\ Michel Marcus, Oct 21 2016

seq(N) = {
  my(bag = List(), cnt = 0, n = 1);
  while(cnt < N,
        if (hammingweight(n)%2 == 0 && hammingweight(n+1) > 1,
            listput(bag, n); cnt++);
        n += 2);
  return(Vec(bag));
};
seq(57)  \\ Gheorghe Coserea, Oct 25 2016

a(2^n - floor(n/2)) = 4*2^n + 1, for all n >= 0. - Gheorghe Coserea, Oct 24 2016

A377563 Numbers that have fewer infinitary divisors than noninfinitary divisors.

Original entry on oeis.org

16, 36, 48, 80, 81, 100, 112, 144, 162, 176, 180, 196, 208, 225, 240, 252, 256, 272, 288, 300, 304, 324, 336, 368, 396, 400, 405, 432, 441, 450, 464, 468, 484, 496, 512, 528, 560, 567, 576, 588, 592, 612, 624, 625, 648, 656, 676, 684, 688, 700, 720, 752, 768, 784, 800

Offset: 1

Amiram Eldar, Nov 01 2024

Numbers whose prime factorization has at least one exponent that has at least two zeros in its binary representation (A158582), or at least two exponents that are not of the form 2^k-1, with k >= 1 (A062289).

The asymptotic density of this sequence is 1 - d * (1 + Sum_{p prime} (Sum_{k>=0} 1/p^(3*2^k-1))/(1 + Sum_{k>=1} 1/p^(2^k-1))) = 0.07306380398261191432..., where d = A327839.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A037445, A062289, A077609, A083329, A158582, A327839, A335832, A348341, A377562.

f[p_, e_] := 2^DigitCount[e, 2, 1]/(e + 1); q[1] = False; q[n_] := Times @@ f @@@ FactorInteger[n] < 1/2; Select[Range[800], q]

is(n) = {my(f = factor(n)); vecprod(apply(x -> (1 << hammingweight(x)) / (x+1), f[, 2])) < 1/2;}

A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 5, 4, 5, 8, 7, 6, 5, 6, 10, 9, 11, 8, 6, 7, 12, 11, 12, 12, 9, 7, 8, 14, 13, 13, 16, 13, 10, 8, 9, 16, 15, 14, 20, 17, 14, 11, 9, 10, 18, 17, 23, 22, 21, 18, 15, 12, 10, 11, 20, 19, 24, 28, 24, 22, 19, 19, 13, 11, 12, 22, 21, 25, 32, 29

Offset: 1

Rémy Sigrist, Mar 08 2018

Each positive number k appears A122953(k) times in this array.

			Square array begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    2    3    4    5    6    7    8    9   10  <--  A000027
    2|    2    4    5    6    8    9   10   11   12   13  <--  A062289
    3|    3    6    7   11   12   13   14   15   19   22  <--  A004780
    4|    4    8    9   12   16   17   18   19   20   24  <--  A004753
    5|    5   10   11   13   20   21   22   23   26   27  <--  A004748
    6|    6   12   13   14   22   24   25   26   27   28  <--  A004749
    7|    7   14   15   23   28   29   30   31   39   46  <--  A004781
    8|    8   16   17   24   32   33   34   35   40   48  <--  A004779
    9|    9   18   19   25   36   37   38   39   41   50
   10|   10   20   21   26   40   41   42   43   52   53  <--  A132782

Rémy Sigrist, Perl program for A300302

Cf. A000027, A004748, A004749, A004753, A004779, A004780, A004781, A062289, A122953, A132782.

Perl
```
See Links section.
```

T(n, 1) = n.
T(n, 2) = 2*n.
T(n, 3) = 2*n + 1.
T(1, n) = A000027(n).
T(2, n) = A062289(n).
T(3, n) = A004780(n).
T(4, n) = A004753(n).
T(5, n) = A004748(n).
T(6, n) = A004749(n).
T(7, n) = A004781(n).
T(8, n) = A004779(n-1).
T(10, n) = A132782(n).

A329367 Numbers whose binary expansion, without the most significant digit, is not a necklace.

Original entry on oeis.org

6, 10, 12, 13, 14, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 34, 36, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Nov 15 2019

nonn

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their binary expansions begins:
   6: (1,1,0)
  10: (1,0,1,0)
  12: (1,1,0,0)
  13: (1,1,0,1)
  14: (1,1,1,0)
  18: (1,0,0,1,0)
  20: (1,0,1,0,0)
  22: (1,0,1,1,0)
  24: (1,1,0,0,0)
  25: (1,1,0,0,1)
  26: (1,1,0,1,0)
  27: (1,1,0,1,1)
  28: (1,1,1,0,0)
  29: (1,1,1,0,1)
  30: (1,1,1,1,0)
  34: (1,0,0,0,1,0)
  36: (1,0,0,1,0,0)
  38: (1,0,0,1,1,0)
  40: (1,0,1,0,0,0)
  41: (1,0,1,0,0,1)

The complement is A328668.
The version involving all digits is A062289.
The reverse version is A328607.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose binary expansion is a necklace are A275692.
Numbers whose reversed binary expansion is a necklace are A328595.
Cf. A000120, A001037, A003714, A065609, A069010, A257739, A328596.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],!neckQ[Rest[IntegerDigits[#,2]]]&]

cp's OEIS Frontend

A138890 Non-Padovan numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

Extensions

A138891 Non-Motzkin numbers.

Views

Author

Keywords

Comments

Crossrefs

A138887 Numbers that are not Sophie Germain primes.

Views

Author

Keywords

Comments

Crossrefs

A257130 Where the difference A055938(n) - A005187(n) obtains record values; positions of records in A257126.

Views

Author

Keywords

Comments

Crossrefs

A276194 Odd numbers whose binary representation contains an even number of 1's and at least one 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A377563 Numbers that have fewer infinitary divisors than noninfinitary divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Perl

Formula

A329367 Numbers whose binary expansion, without the most significant digit, is not a necklace.

Views

Author

Keywords

Comments

Examples