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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A352922 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives m(n).

Original entry on oeis.org

0, 1, 4, 3, 6, 6, 8, 8, 10, 10, 11, 14, 14, 16, 18, 18, 18, 20
Offset: 1

Views

Author

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 24 2022)

Keywords

Comments

The sequences m, p, r are well-defined since every number appears in A109812, and if A109812(s) = 2^n - 1, then by definition both A109812(s-1) and A109812(s+1) must be multiples of 2^n.
The sequences m, p, r are discussed in A352920.
(We assume A109812(0)=0 in order to get m(1)=0.)

Crossrefs

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352792 a(n) is the number of numbers k < n such that A109812(k) < A109812(n).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 7, 5, 7, 10, 9, 11, 10, 14, 6, 15, 17, 15, 10, 20, 16, 22, 18, 22, 12, 26, 14, 28, 21, 28, 31, 23, 28, 33, 24, 29, 37, 22, 39, 26, 39, 30, 27, 41, 37, 45, 35, 47, 49, 28, 50, 52, 29, 51, 55, 41, 50, 58, 40, 49, 44, 52, 45, 54, 44, 53, 47
Offset: 1

Views

Author

Rémy Sigrist, Apr 03 2022

Keywords

Examples

			The initial values of a(n), b(n) = A109812(n), and the corresponding k's, are:
  n   a(n)  b(n)  k's
  --  ----  ----  ---------------------------------------------------
   1     0     1  []
   2     1     2  [1]
   3     2     4  [1, 2]
   4     2     3  [1, 2]
   5     4     8  [1, 2, 3, 4]
   6     4     5  [1, 2, 3, 4]
   7     6    10  [1, 2, 3, 4, 5, 6]
   8     7    16  [1, 2, 3, 4, 5, 6, 7]
   9     5     6  [1, 2, 3, 4, 6]
  10     7     9  [1, 2, 3, 4, 5, 6, 9]
  11    10    18  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
  12     9    12  [1, 2, 3, 4, 5, 6, 7, 9, 10]
  13    11    17  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12]
  14    10    14  [1, 2, 3, 4, 5, 6, 7, 9, 10, 12]
  15    14    32  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
  16     6     7  [1, 2, 3, 4, 6, 9]
  17    15    24  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16]

Links

Crossrefs

Cf. A109812, A352204.

Formula

a(n) <= n-1 with equality iff n belongs to A352204.

A352918 Values of A109812(k) where A109812(k)/k reaches a new high point.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 96, 128, 320, 512, 2048, 2304, 19922944, 41943040, 167772160
Offset: 1

Views

Author

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 21 2022)

Keywords

Comments

The corresponding values of k are given in A352917.
This is a subset of A352203.
The slow growth of A109812(k)/k (see Examples section) suggests that A109812(k)/k is bounded. That is, it appears there is a constant c (between 3.7 and 4) such that A109812(k) < c*k for all k.

Examples

			Let c(k) denote A109812(k). The first 15 record high-points of c(k)/k are as follows:
[c(k)/k, k, c(k), "binary(c(n))"]
[1.000000000, 1, 1, "1"]
[1.333333333, 3, 4, "100"]
[1.600000000, 5, 8, "1000"]
[2.000000000, 8, 16, "10000"]
[2.133333333, 15, 32, "100000"]
[2.206896552, 29, 64, "1000000"]
[2.400000000, 40, 96, "1100000"]
[2.560000000, 50, 128, "10000000"]
[2.962962963, 108, 320, "101000000"]
[3.121951220, 164, 512, "1000000000"]
[3.155624037, 649, 2048, "100000000000"]
[3.539170507, 651, 2304, "100100000000"]
[3.616182275, 5509386, 19922944, "1001100000000000000000000"]
[3.721304271, 11271059, 41943040, "10100000000000000000000000"]
[3.727433952, 45010096, 167772160, "1010000000000000000000000000"]
The values of k and c(k) form A352917 and the present sequence.

Crossrefs

Cf. A109812, A352203, A352204, A352917.

A352919 Indices k where k/A109812(k) reaches a new high point.

Original entry on oeis.org

1, 4, 9, 16, 76, 162, 418, 1892, 19094, 19298, 20059, 84653, 174566, 1688099
Offset: 1

Views

Author

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 23 2022)

Keywords

Comments

The corresponding values of A109812(k) are given in A352920.
This is a subset of A352359.

Examples

			Let c(k) denote A109812(k). The first 14 record high-points of k/c(k) are as follows:
[k/c(k), k, c(k), "binary(c(n))"]
[1.000000000 1 1 "1"]
[1.333333333 4 3 "11"]
[1.500000000 9 6 "110"]
[2.285714286 16 7 "111"]
[2.451612903 76 31 "11111"]
[2.571428571 162 63 "111111"]
[3.291338583 418 127 "1111111"]
[3.702544031 1892 511 "111111111"]
[4.665037870 19094 4093 "111111111101"]
[4.713727406 19298 4094 "111111111110"]
[4.898412698 20059 4095 "111111111111"]
[5.167124458 84653 16383 "11111111111111"]
[5.327494125 174566 32767 "111111111111111"]
[6.439611205 1688099 262143 "111111111111111111"]
The values of k and c(k) form the present sequence and A352920.

Crossrefs

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.

A352921 Let c(s) denote A109812(s). Suppose c(s) = 2^n - 1, and define m(n), p(n), r(n) by m(n) = c(s-1)/2^n, p(n) = c(s+1)/2^n, r(n) = max(m(n), p(n)); sequence gives p(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 9, 9, 11, 12, 13, 13, 15, 15, 17, 17, 19
Offset: 1

Views

Author

David Broadhurst, Aug 17 2022 (entry created by N. J. A. Sloane, Apr 24 2022)

Keywords

Comments

The sequences m, p, r are well-defined since every number appears in A109812, and if A109812(s) = 2^n - 1, then by definition both A109812(s-1) and A109812(s+1) must be multiples of 2^n.
The sequences m, p, r are discussed in A352920.

Crossrefs

Cf. A109812, A113233, A352203, A352204, A352336, A352359, A352917-A352923.
Previous Showing 11-15 of 15 results.

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