A352203 -id:A352203 - cp's OEIS frontend

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).

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

Offset: 1

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

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.)

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

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

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

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.

			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.

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

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

The corresponding values of A109812(k) are given in A352920.

This is a subset of A352359.

			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.

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

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

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.

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

cp's OEIS Frontend

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).

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

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

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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).

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