A301983 -id:A301983 - cp's OEIS frontend

A303077 a(1) = 1, and for n > 1, a(n) is the greatest prime number whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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

Offset: 1

Rémy Sigrist, Apr 18 2018

This sequence has similarities with A078833; there binary digits have to be consecutive, here not.

For n > 1, a(n) is the greatest prime number appearing in the n-th row of A301983.

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   1     1       1       1
   2     2      10      10
   3     3      11      11
   4     2     100     10_
   5     5     101     101
   6     3     110     11_
   7     7     111     111
   8     2    1000    10__
   9     5    1001    10_1
  10     5    1010    101_
  11    11    1011    1011
  12     3    1100    11__
  13    13    1101    1101
  14     7    1110    111_
  15     7    1111    111_

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

Cf. A078833, A301983.

a(n) = my (s=Set(1), b=binary(n)); for (i=2, #b, s=setunion(s, Set(apply(k->2*k+b[i], s)))); vecmax(select(k->k==1 || isprime(k), s))

a(2*n) = a(n) for any n > 1.
a(n) = n iff n is not composite.
a(n) = 2 iff n = 2^k for some k > 0.
a(n) >= A078833(n).

A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 1, 2, 5, 6, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 7, 8, 7, 7, 3, 4, 9, 10, 7, 8, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 11, 12, 11, 11, 5, 6, 13, 14

Offset: 1

Rémy Sigrist, Mar 30 2018

Equivalently, a(n) is the greatest positive number k such that A301983(n, k) = k.

Apparently, the k-th record value is A089633(k), and the first term with this value has index A048678(A089633(k)).

			The 13th row of A301983 is: 1, 2, 3, 5, 6, 7, 13; all numbers in the range 1..3 appear in this row, but the number 4 is missing; hence a(13) = 3.

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

Cf. A048678, A089633, A261461, A301977, A301983.

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

a(n) <= A301977(n).
a(2*n) >= a(n).
a(2*n + 1) >= a(n) (with strict inequality if a(n) is even).
a(n) = 1 iff n is positive and belongs to A000225.

A332030 a(n) is the product of the 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, 1, 2, 3, 8, 30, 36, 21, 64, 1080, 7200, 2310, 1728, 16380, 3528, 315, 1024, 146880, 9331200, 1580040, 13824000, 1362160800, 170755200, 796950, 331776, 176904000, 2861913600, 72972900, 4741632, 99754200, 1587600, 9765, 32768, 77552640, 86294937600

Offset: 0

Rémy Sigrist, Feb 05 2020

This sequence is a variant of A165153.

For n > 0, a(n) is the product of the terms of the n-th row of A301983.

			For n = 9:
- the binary representation of 9 is "1001",
- the following positive binary strings appear in it: "1", "10", "11", "100", "101" and "1001",
- they correspond to: 1, 2, 3, 4, 5 and 9,
- so a(9) = 1 * 2 * 3 * 4 * 5 * 9 = 1080.

Rémy Sigrist, Table of n, a(n) for n = 0..4096

Cf. A005329, A006125, A165153, A301983, A328379 (additive variant).

a(n) = my (b=binary(n), s=[0]); for (i=1, #b, s=setunion(s, apply(m -> 2*m+b[i], s))); vecprod(s[2..#s])

a(n) >= A165153(n).
a(2^k) = A006125(k+1) for any k >= 0.
a(2^k-1) = A005329(k) for any k >= 0.

A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 2, 4, 8, 11, 8, 8, 11, 8, 4, 8, 20, 34, 26, 34, 51, 40, 20, 20, 40, 51, 34, 26, 34, 20, 8, 16, 48, 96, 76, 118, 186, 152, 76, 96, 208, 281, 186, 152, 208, 124, 48, 48, 124, 208, 152, 186, 281, 208, 96, 76, 152, 186, 118, 76, 96, 48, 16, 32

Offset: 1

Rémy Sigrist, Feb 02 2023

This sequence is a variant of A165418.

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------------------
   1     1  N/A
   2     1  {1}
   3     1  {1}
   4     2  {1, 2}
   5     3  {1, 2, 3}
   6     3  {1, 2, 3}
   7     2  {1, 3}
   8     4  {1, 2, 4}
   9     8  {1, 2, 3, 4, 5}
  10    11  {1, 2, 3, 4, 5, 6}
  11     8  {1, 2, 3, 5, 7}
  12     8  {1, 2, 3, 4, 6}
  13    11  {1, 2, 3, 5, 6, 7}
  14     8  {1, 2, 3, 6, 7}
  15     4  {1, 3, 7}
  16     8  {1, 2, 4, 8}

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A165418, A301983.

{ for (n=1, #a=vector(64), print1 (a[n]=if (n==1, 1, s = [1]; b = binary(n); for (k=2, #b, s = setunion(s, apply(v -> 2*v+b[k], s))); sum(k=1, #s-1, a[s[k]]);)", ")) }

a(n) = Sum_{k = 1..A301977(n-1)} a(A301983(n, k)) for any n > 1.
a(2^k) = 2^(k-1) for any k > 0.
a(2^k-1) = 2^(k-2) for any k > 1.
a(n) >= A165418(n).

A363164 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the greatest nonnegative number whose binary digits appear in order but not necessarily as consecutive digits in the binary expansions of n and k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 1, 3, 2, 2, 3, 1, 1, 0, 0, 1, 2, 3, 2, 5, 2, 3, 2, 1, 0, 0, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 0, 0, 1, 2, 3, 4, 3, 6, 3, 4, 3, 2, 1, 0

Offset: 0

Rémy Sigrist, Jul 07 2023

			Table A(n, k) begins:
  n\k | 0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ----+-----------------------------------------------------
    0 | 0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
    1 | 0  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1
    2 | 0  1  2  1  2  2  2  1  2  2   2   2   2   2   2   1
    3 | 0  1  1  3  1  3  3  3  1  3   3   3   3   3   3   3
    4 | 0  1  2  1  4  2  2  1  4  4   4   2   4   2   2   1
    5 | 0  1  2  3  2  5  3  3  2  5   5   5   3   5   3   3
    6 | 0  1  2  3  2  3  6  3  2  3   6   3   6   6   6   3
    7 | 0  1  1  3  1  3  3  7  1  3   3   7   3   7   7   7
    8 | 0  1  2  1  4  2  2  1  8  4   4   2   4   2   2   1
    9 | 0  1  2  3  4  5  3  3  4  9   5   5   4   5   3   3
   10 | 0  1  2  3  4  5  6  3  4  5  10   5   6   6   6   3
   11 | 0  1  2  3  2  5  3  7  2  5   5  11   3   7   7   7
   12 | 0  1  2  3  4  3  6  3  4  4   6   3  12   6   6   3
   13 | 0  1  2  3  2  5  6  7  2  5   6   7   6  13   7   7
   14 | 0  1  2  3  2  3  6  7  2  3   6   7   6   7  14   7
   15 | 0  1  1  3  1  3  3  7  1  3   3   7   3   7   7  15

Rémy Sigrist, Colored representation of the array for n, k < 2^10

See A175466 for a similar sequence.

Cf. A301983.

A(n, k) = { my (sn = [0], bn = binary(n), sk = [0], bk = binary(k)); for (i = 1, #bn, sn = setunion(sn, [2*v+bn[i]|v<-sn])); for (i = 1, #bk, sk = setunion(sk, [2*v+bk[i]|v<-sk])); vecmax(setintersect(sn, sk)); }

A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, 1) = 1 for any n > 0.
A(n, n) = n.

cp's OEIS Frontend

A303077 a(1) = 1, and for n > 1, a(n) is the greatest prime number whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

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A332030 a(n) is the product of the 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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A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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A363164 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the greatest nonnegative number whose binary digits appear in order but not necessarily as consecutive digits in the binary expansions of n and k.

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