cp's OEIS Frontend

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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A368609 a(n) = A368945(A071561(n)).

Original entry on oeis.org

0, 2, 4, 0, 8, 10, 2, 14, 16, 0, 6, 20, 8, 2, 26, 28, 4, 12, 34, 14, 6, 38, 40, 2, 18, 44, 10, 4, 50, 0, 12, 24, 56, 58, 26, 2, 64, 8, 16, 68, 70, 32, 4, 10, 0, 76, 36, 80, 6, 38, 22, 86, 14, 24, 42, 8, 94, 98, 4, 100, 0, 48, 104, 106, 30, 110, 6, 12, 20, 54, 2, 56, 34, 22, 14, 124, 36, 128, 4, 62
Offset: 1

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Author

Hartmut F. W. Hoft, Jan 25 2024

Keywords

Comments

This sequence lists all nonnegative numbers in A368945, all of which are even.
Conjecture: Every nonnegative even number occurs in this sequence.

Examples

			a(4) = A368945(A071561(4)) = A368945(10) = 0 and a(5) = A368945(A071561(5)) = A368945(11) = 8.
For numbers k <= 10^6 the largest width 0 extent instantiated is 999980 for prime 999983 and the smallest width 0 extent not instantiated by any k <= 10^6 is 31396.

Crossrefs

Cf. A071561, A071562, A235791, A237048, A237270, A237593, A249223, A368945.

Programs

  • Mathematica
    a071561Q[n_] := Select[Divisors[n], Sqrt[n/2]<=#A249223 *)
zeroExt[n_] := Module[{s=Position[t249223[n], 1][[-1, -1]]}, 2 Ceiling[(n+1)/(s+1)-(s+1)/2]-2]
a368609[n_] := Map[zeroExt, Select[Range[n], a071561Q]]
a368609[135]

A370204 a(n) is the smallest number k for which the length of the central extent of width 0 in the symmetric representation of sigma, SRS(k), equals 2*n and is -1 if there is no such extent of length 2*n.

Original entry on oeis.org

3, 5, 7, 22, 11, 13, 34, 17, 19, 46, 23, 87, 58, 29, 31, 111, 74, 37, 82, 41, 43, 94, 47, 159, 106, 53, 177, 118, 59, 61, 201, 134, 67, 142, 71, 73, 237, 158, 79, 166, 83, 267, 178, 89, 388, 291, 194, 97, 202, 101, 103, 214, 107, 109, 226, 113, 889, 762, 635, 508, 381, 254, 127, 262, 131, 411
Offset: 0

Views

Author

Hartmut F. W. Hoft, Feb 11 2024

Keywords

Comments

Indices of the first occurrence of value 2*n in A368945.
SRS(a(n)) has an even number of parts.
The maximum possible central 0 width extent in SRS(n) for odd numbers n is 2*n - (n+1) - 2 = n - 3. This is achieved only by odd prime numbers which form a subsequence.
Conjecture: a(n) != -1 for all n >= 0.

Examples

			a(2) = 7 since prime 7 is the smallest number whose central extent of width 0 equals 4.
a(3) = 22 since 22 is the smallest number whose central extent of width 0 equals 6.

Crossrefs

Cf. A000040, A071561, A071562, A086801, A100484, A154115, A235791, A237048, A237270, A237593, A249223, A368945.

Programs

  • Mathematica
    (* Function extent0[ ] is defined in A368945 *)
smallest[n_] := NestWhile[#+1&, n, extent0[#]!=n&]/;EvenQ[n]
a370204[n_] := Map[smallest[2#]&, Range[0, n]]
a370204[65]

Formula

a(n) = min( k : A368945(k) = 2*n ), 0<=n, if the minimum exists, a(n) = -1 otherwise.
A368945(a(k)) = 2 * k, k>=0 and a(k) != -1.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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