A070939 -id:A070939 - cp's OEIS frontend

A232235 Primes that can be written in binary representation as concatenation of two primes: p and p-2. That is, primes representable as p * 2^L + p-2, where p and p-2 are primes, and L is the length of binary representation of p-2: L = A070939(p-2).

23, 61, 1021, 139021, 145177, 222127, 2645257, 2706727, 2928019, 3050959, 3997597, 38695537, 45086077, 49903561, 50247667, 53688727, 56294101, 545636617, 556450387, 558023299, 563331877, 563921719, 581616979, 582993277, 607570027, 619956709, 638045197, 660262579

Offset: 1

Alex Ratushnyak, Nov 20 2013

A subsequence of A232085.

Cf. A000040, A070939, A232085.

a(n) = A232237(n) * 2^A070939(A232237(n)-2) + A232237(n)-2.

A232236 Primes that can be written in binary representation as concatenation of two primes: p and p+2. That is, primes representable as p * 2^L + p+2, where p and p+2 are primes, and L is the length of binary representation of p+2: L = A070939(p+2).

Original entry on oeis.org

29, 47, 563, 9161, 137999, 2149403, 2358401, 3526331, 41776109, 43250849, 46347803, 51607709, 53819819, 55540349, 59866253, 62176679, 539082821, 545571083, 546947381, 625199753, 627165893, 629525261, 650169731, 654102011, 680644901, 687526391, 688509461, 690082373

Offset: 1

Alex Ratushnyak, Nov 20 2013

A subsequence of A232085.

Cf. A000040, A070939, A232085.

a(n) = A232238(n) * 2^A070939(A232238(n)+2) + A232238(n)+2.

A232239 Lesser of twin-bin primes: primes p such that p+2, x and y are primes, where x is concatenation of binary representations of p and p+2, and y is concatenation of binary representations of p+2 and p: x = p * 2^A070939(p+2) + p+2, y = (p+2) * 2^A070939(p) + p.

Original entry on oeis.org

3, 5, 269, 16649, 27689, 29129, 82889, 93239, 129629, 274199, 289169, 309479, 336899, 349079, 371339, 374639, 415109, 454709, 463889, 492719, 1051079, 1063919, 1127309, 1198289, 1209779, 1229519, 1268789, 1350959, 1354649, 1355279, 1392539, 1430879, 1547129, 1551959

Offset: 1

Alex Ratushnyak, Nov 20 2013

Conjecture: the sequence is infinite.

			269 is in the sequence because the following are three primes: 271, 269 * 512 + 271 = 137999, 271 * 512 + 269 = 139021.

Cf. A001359, A070939, A232237, A232238.

import java.math.BigInteger;
public class A232239 {
public static void main (String[] args) {
long bl = 2, next = 3; // bit length, next n such that bl++ for n + 2
for (long n = 3; n < 0xffffffffL; n += 2) {
  long blPrev = bl;
  if (n == next) { ++bl; next = next * 2 + 1; }
  if (BigInteger.valueOf(n).isProbablePrime(80) &&
    BigInteger.valueOf(n + 2).isProbablePrime(80) &&
    BigInteger.valueOf((n << bl) + n + 2).isProbablePrime(80) &&
    BigInteger.valueOf(((n + 2) << blPrev) + n).isProbablePrime(80))
        System.out.printf("%d, ", n);
}
}
}

Select[Prime[Range[200]], PrimeQ[# + 2] && PrimeQ[FromDigits[Flatten[{IntegerDigits[#, 2], IntegerDigits[# + 2, 2]}], 2]] && PrimeQ[FromDigits[Flatten[{IntegerDigits[# + 2, 2], IntegerDigits[#, 2]}], 2]] &] (* Alonso del Arte, Jan 19 2014 *)

A264749 a(n) = floor(n/BL(n)) where BL(n) = A070939(n) is the binary length of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 0

Alex Ratushnyak, Nov 23 2015

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

Cf. A070939, A135941, A253608.

a264749 n = div n $ a070939 n  -- Reinhard Zumkeller, Dec 05 2015

{0}~Join~Table[Floor[n/IntegerLength[n, 2]], {n, 84}] (* Michael De Vlieger, Dec 01 2015 *)

a(n) = if (n, n\#binary(n)); \\ Michel Marcus, Dec 01 2015

for n in range(88):  print(n // (len(bin(n))-2), end=', ')

A318687 Number of length-n circular binary words having exactly n distinct blocks of length floor(log_2(n)) + 1 (A070939).

Original entry on oeis.org

2, 1, 2, 3, 2, 3, 4, 12, 14, 17, 14, 13, 12, 20, 32, 406, 538, 703, 842, 1085, 1310, 1465, 1544, 1570, 1968, 2132, 2000, 2480, 2176, 2816, 4096, 1060280

Offset: 1

Jeffrey Shallit, Aug 31 2018

A "circular word" (a.k.a. "necklace") is one that wraps around from the end to the beginning. The words are counted up to an equivalence where two circular words are the same if one is a cyclic shift of the other.

D. Gabric, S. Holub, and J. Shallit, Generalized de Bruijn words and the state complexity of conjugate sets, arXiv:1903.05442 [cs.FL], March 13 2019.

Cf. A317586, which studies a similar quantity for two different lengths of blocks.

Cf. A070939.

a(2^n-1) = 2^(2^(n-1)-n+1) since A317586(2^n) = 2^(2^(n-1)-n) and A317586(2^n-1) = A317586(2^n+1) = 2*A317586(2^n) = 2^(2^(n-1)-n+1). - Altug Alkan, Sep 05 2018

A320675 Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [gcd(m, i)=1] for i = 1..k (where [] is an Iverson bracket).

Original entry on oeis.org

1, 2, 3, 7, 10, 20, 27, 31, 40, 127, 138, 219, 245, 276, 552, 650, 682, 1364, 2047, 2728, 8191, 10922, 13515, 14043, 32747, 112347, 131071, 524287, 2796202, 3459945, 5592404, 7124187, 8388607, 8530050, 10660010, 11184808, 16645111, 17060100, 21320020, 33554431

Offset: 1

Rémy Sigrist, Oct 19 2018

In other words, the ones in the binary representation of a term of this sequence encode the first numbers coprime to this term.
This sequence contains every term of A001348: 2^2 - 1 belongs to this sequence, and for any odd prime number p, if q divides 2^p - 1, then q > p and gcd(p, i) = 1 for i = 1..p.
See A320673 for similar sequences.

			The first terms, alongside their binary representation and the coprime numbers encoded therein, are:
  n   a(n)  bin(a(n))  First numbers coprime
  --  ----  ---------  ---------------------
   1     1  1          1
   2     2  10         1
   3     3  11         1, 2
   4     7  111        1, 2, 3
   5    10  1010       1, 3
   6    20  10100      1, 3
   7    27  11011      1, 2, 4, 5
   8    31  11111      1, 2, 3, 4, 5
   9    40  101000     1, 3
  10   127  1111111    1, 2, 3, 4, 5, 6, 7

Cf. A001348, A070939, A320673.

is(n) = my (b=binary(n)); b==vector(#b, k, gcd(n, k)==1)

A324391 Fully multiplicative with a(p^e) = A070939(p)^e, where A070939(p) gives the length of the binary representation of p.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 4, 8, 4, 6, 6, 16, 5, 8, 5, 12, 6, 8, 5, 16, 9, 8, 8, 12, 5, 12, 5, 32, 8, 10, 9, 16, 6, 10, 8, 24, 6, 12, 6, 16, 12, 10, 6, 32, 9, 18, 10, 16, 6, 16, 12, 24, 10, 10, 6, 24, 6, 10, 12, 64, 12, 16, 7, 20, 10, 18, 7, 32, 7, 12, 18, 20, 12, 16, 7, 48, 16, 12, 7, 24, 15, 12, 10, 32, 7, 24, 12, 20, 10, 12

Offset: 1

Antti Karttunen, Mar 05 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000043, A000668, A070939, A072084.

A070939(n) = if(!n,1,#binary(n));
A324391(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = A070939(f[i, 1])); factorback(f); };

For all n >= 1, a(A000668(n)) = A000043(n).

A365746 Table read by antidiagonals upward: T(n,k) is the number of binary strings of length k with the property that every substring of length A070939(n) is lexicographically earlier than the binary expansion of n; n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 5, 2, 1, 0, 1, 2, 4, 4, 8, 2, 1, 0, 1, 2, 4, 5, 4, 13, 2, 1, 0, 1, 2, 4, 6, 7, 4, 21, 2, 1, 0, 1, 2, 4, 7, 10, 11, 4, 34, 2, 1, 0, 1, 2, 4, 8, 13, 16, 16, 4, 55, 2, 1, 0, 1, 2, 4, 8, 8, 24

Offset: 0

Peter Kagey, Sep 17 2023

			Table begins:
 n\k | 0  1  2  3   4   5   6   7    8    9   10   11
-----+----------------------------------------------------
   0 | 1, 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0, ...
   1 | 1, 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1, ...
   2 | 1, 2, 2, 2,  2,  2,  2,  2,   2,   2,   2,   2, ...
   3 | 1, 2, 3, 5,  8, 13, 21, 34,  55,  89, 144, 233, ...
   4 | 1, 2, 4, 4,  4,  4,  4,  4,   4,   4,   4,   4, ...
   5 | 1, 2, 4, 5,  7, 11, 16, 23,  34,  50,  73, 107, ...
   6 | 1, 2, 4, 6, 10, 16, 26, 42,  68, 110, 178, 288, ...
   7 | 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, ...
   8 | 1, 2, 4, 8,  8,  8,  8,  8,   8,   8,   8,   8, ...
   9 | 1, 2, 4, 8,  9, 11, 15, 23,  32,  43,  58,  81, ...
For (n,k) = (3,4), we see that T(3,4) = 8 because there are 8 binary strings of length k = 4 where all length A070939(3) = 2 substrings are lexicographically earlier than "11" (the binary expansion of n = 3): 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010.

Cf. A030190, A070939, A209972.

Cf. A000045 (row 3), A164316 (row 5), A128588 (row 6), A000073 (row 7).

A365746Row[s_,
  numberOfTerms_] := (digits = If[s == 0, 1, Ceiling[Log[2, s + 1]]];
  m = 2^(digits - 1);
  transferMatrix =
   If[s == 0, {{0}},
    Table[If[(Ceiling[i/2] ==
         j) || ((i <= s - m) && (Ceiling[i/2] == j - m/2)), 1, 0], {i,
       1, m}, {j, 1, m}]];
  sequence =
   Table[2^k, {k, 0, digits - 1}] ~Join~
    Table[MatrixPower[transferMatrix, k] // Total // Total, {k, 1,
      numberOfTerms - digits}];
  Take[sequence, numberOfTerms])

G.f. for row n = 0: 1;
G.f. for row n = 1: 1/(1 - x);
G.f. for row n = 2: (1 + x)/(1 - x);
G.f. for row n = 3: (1 + x)/(1 - x - x^2);
G.f. for row n = 4: (1 + x + 2x^2)/(1 - x);
G.f. for row n = 5: (1 + x + 2x^2)/(1 - x - x^3);
G.f. for row n = 6: (1 + x + x^2)/(1 - x - x^2);
G.f. for row n = 7: (1 + x + x^2)/(1 - x - x^2 - x^3);
G.f. for row n = 8: (1 + x + 2 x^2 + 4 x^3)/(1 - x);
G.f. for row n = 9: (1 + x + 2x^2 + 4x^3)/(1 - x - x^4).

A371343 Lexicographically latest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and of a(n) have the same length (A070939) and the same number of runs of consecutive equals digits (A005811).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 14, 13, 10, 11, 12, 9, 8, 15, 30, 29, 26, 27, 22, 21, 20, 25, 28, 23, 18, 19, 24, 17, 16, 31, 62, 61, 58, 59, 54, 53, 52, 57, 50, 45, 42, 43, 46, 41, 44, 55, 60, 51, 40, 49, 38, 37, 36, 47, 56, 39, 34, 35, 48, 33, 32, 63, 126, 125, 122

Offset: 0

Rémy Sigrist, Mar 24 2024

This sequence is a self-inverse permutation of the nonnegative integers with infinitely many fixed points (for example, all terms of A000225 are fixed points).

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0
   1     1       1          1
   2     2      10         10
   3     3      11         11
   4     6     100        110
   5     5     101        101
   6     4     110        100
   7     7     111        111
   8    14    1000       1110
   9    13    1001       1101
  10    10    1010       1010
  11    11    1011       1011
  12    12    1100       1100
  13     9    1101       1001
  14     8    1110       1000
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^20 terms (where the color is function of A005811(n))
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

See A331274 and A337242 for similar sequences.

Cf. A000225, A005811, A070939.

PARI
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\\ See Links section.
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A380294 The Golomb-Rice encoding of n, with M = A070939(A070939(n)).

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 48, 49, 50, 51, 52, 53, 54, 55, 112, 113, 114, 115, 116, 117, 118, 119, 240, 241, 242, 243, 244, 245, 246, 247, 496, 497, 498, 499, 500, 501, 502, 503, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 2032

Offset: 1

Darío Clavijo, Jan 19 2025

For n <= 3 leading zeros are omitted.
In the original Golomb-Rice algorithm, M is an arbitrary power of 2. Here, M is determined as 2^(floor(log_2(floor(log_2(n))+1))+1).
a(2^k) is a divisor of a(2^(k+1)).

			For n = 42 a(42) = 111110010_2 = 498.
m = 2^(floor(log_2(floor(log_2(n))+1))+1) = 8 and
q = floor(n / m) = 5 and
r = n mod m = 2 and
u = (2^q-1) left shifted by (2^floor(log_2(m)) + 1) = 111110000_2 = 496 and
u + r = 111110010_2 = 498.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Golomb coding

Cf. A000079, A000225, A070939.

A380294[n_] := (2^Quotient[n, #] - 1)*2^BitLength[#] + Mod[n, #] & [2^Nest[BitLength, n, 2]];
Array[A380294, 100] (* Paolo Xausa, Feb 03 2025 *)

def a(n):
    if n <= 3: return n
    if n & 1: return a(n-1)+1
    m = 1 << (n.bit_length()).bit_length()
    q, r = divmod(n, m)
    u = ((1 << q) - 1) << m.bit_length()
    return u + r
print([a(n) for n in range(1,57)])

a(n) = (2^q - 1) * 2^A070939(m) + r, where m = 2^A070939(A070939(n)), q = floor(n/m) and r = n mod m.
a(n) = a(n-1)+1 for n odd > 1.
a(n) = n if n <= 3.

cp's OEIS Frontend

A232235 Primes that can be written in binary representation as concatenation of two primes: p and p-2. That is, primes representable as p * 2^L + p-2, where p and p-2 are primes, and L is the length of binary representation of p-2: L = A070939(p-2).

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A232236 Primes that can be written in binary representation as concatenation of two primes: p and p+2. That is, primes representable as p * 2^L + p+2, where p and p+2 are primes, and L is the length of binary representation of p+2: L = A070939(p+2).

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A232239 Lesser of twin-bin primes: primes p such that p+2, x and y are primes, where x is concatenation of binary representations of p and p+2, and y is concatenation of binary representations of p+2 and p: x = p * 2^A070939(p+2) + p+2, y = (p+2) * 2^A070939(p) + p.

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Java

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A264749 a(n) = floor(n/BL(n)) where BL(n) = A070939(n) is the binary length of n.

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A318687 Number of length-n circular binary words having exactly n distinct blocks of length floor(log_2(n)) + 1 (A070939).

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A320675 Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [gcd(m, i)=1] for i = 1..k (where [] is an Iverson bracket).

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A324391 Fully multiplicative with a(p^e) = A070939(p)^e, where A070939(p) gives the length of the binary representation of p.

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A365746 Table read by antidiagonals upward: T(n,k) is the number of binary strings of length k with the property that every substring of length A070939(n) is lexicographically earlier than the binary expansion of n; n, k >= 0.

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A371343 Lexicographically latest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and of a(n) have the same length (A070939) and the same number of runs of consecutive equals digits (A005811).

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