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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A349811 Table of triples read by rows: the fundamental unit of the purely real cubic field Q(m^(1/3)), m = A349810(n), is (T(n,0) + T(n,1)*x + T(n,2)*x^2)/d, where x = m^(1/3) and d is a small integer, usually 1.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 41, 24, 14, 109, 60, 33, 4, 2, 1, 23, 11, 5, 89, 40, 18, 110, 48, 21, 94, 40, 17, 29, 12, 5, 5401, 2190, 888, 324, 126, 49, 14, 5, 2, 22, 8, 3, 1705, 618, 224, 793, 283, 101, 2166673601, 761875860, 267901370
Offset: 1

Views

Author

N. J. A. Sloane, Dec 22 2021

Keywords

Comments

The denominator d is 1 with the following exceptions: 3 for m=10, 2 for m=12, 3 for m=19, 2 for m=20, and so on.
Wada's table extends to m=249.

Examples

			The table begins as follows (the denominator d has been included if it is not 1, and m = A349810(n)):
n  m  (T(n,0) T(n,1) T(n,2))/d
1  2  1, 1, 1,
2  3  4, 3, 2,
3  5  41, 24, 14,
4  6  109, 60, 33,
5  7  4, 2, 1,
6 10  (23, 11, 5)/3,
7 11  89, 40, 18,
8 12  (110, 48, 21)/2,
9 13  94, 40, 17,
10 14 29, 12, 5,
11 15 5401, 2190, 888,
12 17 324, 126, 49,
13 19 (14, 5, 2)/3,
14 20 (22, 8, 3)/2,
15 21 1705, 618, 224,
16 22 793, 283, 101,
17 23 2166673601, 761875860, 267901370,
...

Links

Crossrefs

Cf. A349810.

A375402 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) have distinct maxima.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89
Offset: 1

Views

Author

Gus Wiseman, Aug 14 2024

Keywords

Comments

First differs from A349810 in lacking 150.
An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
The partitions with these Heinz numbers are those with (1) no part appearing more than twice and (2) the greatest part appearing only once.
Note the prime factors can alternatively be written in weakly decreasing order.
How is does the sequence relate to A317092? - R. J. Mathar, Aug 20 2024

Examples

			The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is not in the sequence.
The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is in the sequence.
The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is not in the sequence.

Crossrefs

For identical instead of distinct we have A065200, complement A065201.
A version for compositions (instead of partitions) is A374767.
Partitions of this type are counted by A375133.
For minima instead of maxima we have A375398, counted by A375134.
The complement for minima is A375399, counted by A375404.
The complement is A375403, counted by A375401.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A046660, A066328, A358830, A374632, A374706, A374768, A375128, A375136, A375396, A375400.

Programs

  • Mathematica
    Select[Range[150],UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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