A111791 -id:A111791 - cp's OEIS frontend

A113198 Tetrahedron T(g, h, q) = number of rotes of weight g, height h, quench q.

1, 1, 2, 2, 4, 2, 2, 8, 8, 1, 8, 12, 4, 4, 28, 16

Offset: 1

Jon Awbrey, Oct 18 2005

T(g, h, q) = |{m : A062537(m) = g, A109301(m) = h, A108352(m) = q}|.

This is the column that is labeled "r" in the tabulation of A113197.

			Table T(g, h, q), omitting empty cells, starts out as follows:
--------+------------------------------------------------------------
g\(h,q) | (0,1) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` (1,0) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` (2,0) (2,2) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` (3,0) (3,2) (3,3) ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (4,0) (4,2) ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (5,2)
========+============================================================
0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
2 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
3 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
4 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 2 ` ` 8 ` ` ` ` ` ` ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` `
--------+------------------------------------------------------------
5 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` `12 ` ` 4 ` ` ` ` ` ` ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` `28 ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `16 `
--------+------------------------------------------------------------
Row sums = A111793. Horizontal section sums = A061396.

J. Awbrey, Riffs and Rotes

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112846, A112868 to A112871, A113197.

A178484 For n=1,2,... list all numbers not occurring earlier which can be written as a product of the first n primes raised to some nonnegative power less than n.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900, 7, 8, 14, 21, 24, 27, 28, 35, 40, 42, 49, 54, 56, 63, 70, 72, 84, 98, 105, 108, 120, 125, 126, 135, 140, 147, 168, 175, 189, 196, 200, 210, 216, 245, 250, 252

Offset: 1

M. F. Hasler, May 31 2010

nonn

A condensed version of sequence A178483.
Every positive integer occurs exactly once in this sequence, but depending on its largest prime factor, it may appear later than much larger numbers. E.g. 7=a(29) appears after a(28)=900, and 11=a(257) appears only after a(256)=9261000.
The first n^n terms are the divisors of n#^(n-1), so any term divisible by the k-th prime must appear later than position (k-1)^(k-1). - Charlie Neder, Mar 08 2019

			n=1 gives a(1) = 1: numbers 2^a with a < 1.
n=2 gives a(2..4) = [2, 3, 6]: numbers 2^a 3^b with a,b < 2.
n=3 gives a(5..28) = [4, 5, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900]: numbers 2^a 3^b 5^c not occurring earlier, with a,b,c < 3.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A178480, A178483, A111791.

DeleteDuplicates@Flatten@Table[Sort[Times @@ (Prime@Range@n^PadLeft[ IntegerDigits[#, n], n]) & /@ (Range[n^n] - 1)], {n, 2, 4}] (* Ivan Neretin, May 02 2019 *)

{ s=0; for( L=1,4, a=[]; forvec( v=vector(L,i,[0,L-1]), bittest(s,t=prod( j=1,L,prime(j)^v[L-j+1] )) & next; s+=1<

A111794 Integers whose rote weight and rote height are equal, sorted by the equated value.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 16, 11, 17, 19, 32, 53, 128, 256, 65536, 31, 59, 67, 131, 241, 719, 1619, 2048, 131072, 524288, 821641, 4294967296, 9007199254740992

Offset: 1

Jon Awbrey, Aug 28 2005

The number of integers m whose rote weight, g(m) = A062537(m) and rote height, h(m) = A109301(m), are both equal to j is 2^(j-1) for j > 0 and 1 for j = 0, as enumerated by the main diagonal of the array shown with sequence A111793.

			Triangle whose j^th row lists the integers m with g(m) = h(m) = j
j | m such that g(m) = h(m) = j
--+-------------------------------------------------------
0 | 1
1 | 2
2 | 3 4
3 | 5 7 8 16
4 | 11 17 19 32 53 128 256 65536
5 | 31 59 67 131 241 719 1619 2048 131072 524288 821641
` | 4294967296 9007199254740992 2^128 2^256 2^65536

J. Awbrey, Riffs and Rotes

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A109300, A109301, A111791, A111792, A111793, A111800.

A112480 Positive integers sorted by rote weight, rote wagage and rote height.

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 7, 8, 16, 6, 13, 23, 25, 27, 49, 64, 81, 512, 11, 17, 19, 32, 53, 128, 256, 65536, 12, 18, 10, 14, 37, 61, 125, 169, 343, 529, 625, 729, 2401, 4096, 19683, 262144, 29, 41, 43, 83, 97, 103, 121, 227, 243, 289, 311, 361, 419, 1024, 2187, 2809, 3671

Offset: 1

Jon Awbrey, Sep 27 2005

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote wayage or root degree is w(m) = omega(m) = A001221(m) and the rote height in gammas is h(m) = A109301(m).

			Table of Primal Functions, Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g w h | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 0 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 1 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 1 2 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 1 2 | 2 | 2 | 2
================================================================
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 1 2 | 1 | ` |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 1 3 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 1 3 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 1 3 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 1 3 | 4 | 5 |
----------------+---------------------------+-------+---+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 2 2 | 1 | 1 | 6
================================================================
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 1 3 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 1 3 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 1 3 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 1 3 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 1 3 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 1 3 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 1 3 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 1 3 | 8 | ` |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 1 4 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 1 4 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 1 4 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 1 4 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 1 4 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 1 4 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 1 4 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 1 4 | 8 |16 |
----------------+---------------------------+-------+---+---+---
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 2 2 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 2 2 | 2 | ` |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 2 3 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 2 3 | 2 | 4 |20
================================================================
12:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `37 | 5 1 3 | ` | ` |
18:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `61 | 5 1 3 | ` | ` |
3:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 125 | 5 1 3 | ` | ` |
6:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 169 | 5 1 3 | ` | ` |
4:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 343 | 5 1 3 | ` | ` |
9:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 529 | 5 1 3 | ` | ` |
3:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 625 | 5 1 3 | ` | ` |
2:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 729 | 5 1 3 | ` | ` |
4:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2401 | 5 1 3 | ` | ` |
1:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `4096 | 5 1 3 | ` | ` |
2:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 19683 | 5 1 3 | ` | ` |
1:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `262144 | 5 1 3 |12 | ` |
----------------+---------------------------+-------+---+---+---
10:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `29 | 5 1 4 | ` | ` |
13:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `41 | 5 1 4 | ` | ` |
14:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `43 | 5 1 4 | ` | ` |
23:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `83 | 5 1 4 | ` | ` |
25:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `97 | 5 1 4 | ` | ` |
27:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 103 | 5 1 4 | ` | ` |
5:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 121 | 5 1 4 | ` | ` |
49:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 227 | 5 1 4 | ` | ` |
2:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 243 | 5 1 4 | ` | ` |
7:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 289 | 5 1 4 | ` | ` |
64:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 311 | 5 1 4 | ` | ` |
8:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 361 | 5 1 4 | ` | ` |
81:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 419 | 5 1 4 | ` | ` |
1:10` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1024 | 5 1 4 | ` | ` |
2:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2187 | 5 1 4 | ` | ` |
16:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2809 | 5 1 4 | ` | ` |
512:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `3671 | 5 1 4 | ` | ` |
2:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `6561 | 5 1 4 | ` | ` |
1:13` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `8192 | 5 1 4 | ` | ` |
1:14` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 16384 | 5 1 4 | ` | ` |
1:23` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` 8388608 | 5 1 4 | ` | ` |
1:25` ` ` ` ` ` | ` ` ` ` ` ` ` ` `33554432 | 5 1 4 | ` | ` |
2:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` `43046721 | 5 1 4 | ` | ` |
1:27` ` ` ` ` ` | ` ` ` ` ` ` ` ` 134217728 | 5 1 4 | ` | ` |
1:49` ` ` ` ` ` | ` ` ` ` ` 562949953421312 | 5 1 4 | ` | ` |
1:64` ` ` ` ` ` | ` ` `18446744073709551616 | 5 1 4 | ` | ` |
1:81` ` ` ` ` ` | 2417851639229258349412352 | 5 1 4 | ` | ` |
1:512 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^512 | 5 1 4 |28 | ` |
----------------+---------------------------+-------+---+---+---
11:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `31 | 5 1 5 | ` | ` |
17:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `59 | 5 1 5 | ` | ` |
19:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `67 | 5 1 5 | ` | ` |
32:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 131 | 5 1 5 | ` | ` |
53:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 241 | 5 1 5 | ` | ` |
128:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 719 | 5 1 5 | ` | ` |
256:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1619 | 5 1 5 | ` | ` |
1:11` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2048 | 5 1 5 | ` | ` |
1:17` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `131072 | 5 1 5 | ` | ` |
1:19` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `524288 | 5 1 5 | ` | ` |
65536:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` `821641 | 5 1 5 | ` | ` |
1:32` ` ` ` ` ` | ` ` ` ` ` ` ` `4294967296 | 5 1 5 | ` | ` |
1:53` ` ` ` ` ` | ` ` ` ` `9007199254740992 | 5 1 5 | ` | ` |
1:128 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^128 | 5 1 5 | ` | ` |
1:256 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^256 | 5 1 5 | ` | ` |
1:65536 ` ` ` ` | ` ` ` ` ` ` ` ` ` 2^65536 | 5 1 5 |16 |56 |
----------------+---------------------------+-------+---+---+---
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `36 | 5 2 2 | 1 | ` |
----------------+---------------------------+-------+---+---+---
2:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `15 | 5 2 3 | ` | ` |
1:2 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `20 | 5 2 3 | ` | ` |
2:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `21 | 5 2 3 | ` | ` |
1:3 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `24 | 5 2 3 | ` | ` |
1:1 6:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `26 | 5 2 3 | ` | ` |
1:2 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `28 | 5 2 3 | ` | ` |
1:1 9:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `46 | 5 2 3 | ` | ` |
1:4 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `48 | 5 2 3 | ` | ` |
1:1 3:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `50 | 5 2 3 | ` | ` |
1:1 2:3 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `54 | 5 2 3 | ` | ` |
1:1 4:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `98 | 5 2 3 | ` | ` |
1:1 2:4 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 162 | 5 2 3 |12 | ` |
----------------+---------------------------+-------+---+---+---
1:1 5:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `22 | 5 2 4 | ` | ` |
1:1 7:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `34 | 5 2 4 | ` | ` |
1:1 8:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `38 | 5 2 4 | ` | ` |
1:1 16:1` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 106 | 5 2 4 | 4 |17 |73
================================================================
a = this sequence
g = rote weight in gammas = A062537
w = rote wayage in gammas = A001221
h = rote height in gammas = A109301
r = number in (g,h,w) set = A112481
s = count in (g, w) class = A111797
t = count in weight class = A061396

J. Awbrey, Riffs and Rotes

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112481.

A112481 Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 8, 8, 2, 2, 12, 28, 16, 1, 12, 4

Offset: 1

Jon Awbrey, Sep 27 2005

T(g, w, h) = |{m : A062537(m) = g, A001221(m) = w, A109301(m) = h}|.
This is the column that is labeled "r" in the tabulation of A112480.
a(n) is a permutation of the elements in A112096.
g = h > 0 implies w = 1 and T(j, 1, j) = 2^(j-1) = A000079(j-1).

			Table T(g, w, h), omitting empty cells, starts out as follows:
--------+-------------------------------------------------------
g\(w,h) | (0,0) (1,1) (1,2) ` ` ` (1,3) ` ` ` (1,4) ` ` ` (1,5)
` ` ` ` | ` ` ` ` ` ` ` ` ` (2,2) ` ` ` (2,3) ` ` ` (2,4) ` ` `
========+=======================================================
0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
2 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
3 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` ` ` 8 ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `12 ` ` ` ` `28 ` ` ` ` `16 `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` `12 ` ` ` ` ` 4 ` ` ` `
--------+-------------------------------------------------------
Row sums = A111797. Horizontal section sums = A061396.

J. Awbrey, Riffs and Rotes

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112480.

A112870 Positive integers sorted by rote height and primal code characteristic.

Original entry on oeis.org

1, 2, 6, 9, 12, 18, 36, 3, 4

Offset: 1

Jon Awbrey, Oct 14 2005

Positive integers m sorted by h(m) = A109301(m) and q(m) = A108352(m).

Using "quench" as a shorter substitute for "primal code characteristic", the rote corresponding to the positive integer m has a quench of q(m) = A108352(m). Numbers with primal code characteristic 0 are "unquenchable".

			Primal Function | Primal Code = a | h q | s | t
----------------+-----------------+-----+---+---
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` 1 | 0 1 | 1 | 1
----------------+-----------------+-----+---+---
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` 2 | 1 0 | 1 | 1
----------------+-----------------+-----+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` 6 | 2 0 | ` |
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` 9 | 2 0 | ` |
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` `12 | 2 0 | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` `18 | 2 0 | ` |
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` `36 | 2 0 | 5 |
----------------+-----------------+-----+---+---
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` 3 | 2 2 | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` 4 | 2 2 | 2 | 7
----------------+-----------------+-----+---+---
a = this sequence
h = rote height in gammas = A109301
q = primal code character = A108352
s = count in (h, q) class = A112871
t = count in height class = A109300

J. Awbrey, Riffs and Rotes

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112846, A112868, A112869, A112871.

A111799 Triangle T(h, w) = number of rotes of height h and wayage w.

Original entry on oeis.org

1, 1, 3, 4, 77

Offset: 1

Jon Awbrey, Sep 01 2005 - Sep 02 2005

T(h, w) = |{positive integers m : A109301(m) = h and A001221(m) = w}|.

Let c(h) = 1 for h = 0 and A050924(h) for h > 0. In other words, c(h) is the sequence [1, A050924] = [1,1,2,9,10^9, ...] that begins with 1 and continues with the terms of A050924. Then the number of nonzero entries in row h is c(h) and their sum is A109300(h). See A111798 for definitions and further details.

			Table T(h, w), omitting zeros, begins as follows:
h\w| 0 ` 1 ` 2 ` 3 ` 4 ` 5 ` 6 ` 7 ` 8 ` 9
---+---------------------------------------
`0 | 1
`1 | ` ` 1
`2 | ` ` 3 ` 4
`3 | ` `77 ` ? ` ? ` ? ` ? ` ? ` ? ` ? ` ?

J. Awbrey, Riffs and Rotes

Cf. A001221, A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A109300, A109301, A111791, A111792, A111793, A111798, A111800.

A113199 Positive integers sorted by rote weight, rote quench and rote height.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 8, 16, 12, 18, 10, 14, 13, 23, 25, 27, 49, 64, 81, 512, 11, 17, 19, 32, 53, 128, 256, 65536, 36, 26, 46, 50, 54, 98, 125, 162, 2401, 22, 34, 38, 106, 15, 21, 37, 61, 169, 343, 529, 625, 729, 4096, 19683, 262144, 29, 41, 43, 83, 97, 103, 121, 227

Offset: 1

Jon Awbrey, Oct 18 2005

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote quench or primal code characteristic is q(m) = A108352(m) and the rote height in gammas is h(m) = A109301(m).

This sequence begins to differ from A113197 at the 40th term, a(40) = 22.

			Primal Functions, Primal Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g q h | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 1 0 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 0 1 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 2 2 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 2 2 | 2 | 2 | 2
================================================================
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 0 2 | ` | ` |
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 0 2 | 2 | 2 |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 2 3 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 2 3 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 2 3 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 2 3 | 4 | 4 | 6
================================================================
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 0 2 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 0 2 | 2 | ` |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 0 3 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 0 3 | 2 | 4 |
----------------+---------------------------+-------+---+---+---
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 2 3 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 2 3 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 2 3 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 2 3 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 2 3 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 2 3 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 2 3 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 2 3 | 8 | ` |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 2 4 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 2 4 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 2 4 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 2 4 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 2 4 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 2 4 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 2 4 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 2 4 | 8 |16 |20
================================================================
a = this sequence
g = rote weight in gammas = A062537
q = primal code character = A108352
h = rote height in gammas = A109301
r = number in (g,q,h) set = A113200
s = count in (g, q) class = A112869
t = count in weight class = A061396

J. Awbrey, Table for Rote Weights 0 to 5
J. Awbrey, Riffs and Rotes

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112868, A112869, A112870, A112871.
Cf. A113197, A113198, A113200.

A113200 Tetrahedron T(g, q, h) = number of rotes of weight g, quench q, height h.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 2, 8, 8, 1, 8, 4, 12, 28, 16, 4

Offset: 1

Jon Awbrey, Oct 18 2005

T(g, q, h) = |{m : A062537(m) = g, A108352(m) = q, A109301(m) = h}|.
This is the column that is labeled "r" in the tabulation of A113199.
a(n) is a permutation of the elements in A113198.

			Table T(g, q, h), omitting empty cells, starts out as follows:
--------+------------------------------------------------------------
g\(q,h) | (1,0) (0,1) (0,2) ` ` ` (0,3) ` ` ` ` ` ` (0,4) ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` (2,2) ` ` ` (2,3) ` ` ` ` ` ` (2,4) (2,5)
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (3,3) ` ` ` ` ` ` ` ` `
========+============================================================
0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
2 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
3 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
4 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` ` ` ` ` ` 8 ` ` ` `
--------+------------------------------------------------------------
5 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` 8 ` ` ` ` ` ` ` ` 4 ` ` ` ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `12 ` ` ` ` ` ` ` `28 ` `16 `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
Row sums = A112869. Horizontal section sums = A061396.

J. Awbrey, Riffs and Rotes

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112868 to A112871, A113197 to A113199.

A178479 For n=0,1,2,... list all numbers not occurring earlier which can be written as product of the first n primes raised to some nonnegative power not exceeding n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36, 5, 8, 10, 15, 20, 24, 25, 27, 30, 40, 45, 50, 54, 60, 72, 75, 90, 100, 108, 120, 125, 135, 150, 180, 200, 216, 225, 250, 270, 300, 360, 375, 450, 500, 540, 600, 675, 750, 900, 1000, 1080, 1125, 1350, 1500, 1800, 2250, 2700, 3000, 3375

Offset: 1

M. F. Hasler, May 31 2010

nonn

Every positive integer occurs exactly once in this sequence, but depending on its largest prime factor, it may appear quite late with respect to larger numbers. E.g. prime(4)=7=a(65) appears after a(4^3)=27000=(2*3*5)^3, prime(5)=11=a(626) appears after a(5^4)=(2*3*5*7)^4=1944810000.

First A000169(n) terms are the divisors of A181555(n), and a(A000169(n))=A181555(n). [From Matthew Vandermast, Oct 31 2010]

			n=0, n=1 and n=2 give a(1)=1 (empty product), a(2)=2=prime(1)^1,
and a(3..9) = 3, 4, 6, 9, 12, 18, 36: numbers 2^a 3^b with a,b <= 2.
n=3 gives a(10..64) = 5, 8, 10, 12, 15, 18...: numbers 2^a 3^b 5^c not occurring earlier, with a,b,c <= 3.

Cf. A178480, A178483, A178484, A111791.

{ s=[]; for( L=0,3, a=[]; forvec( v=vector(L,i,[0,L]), setsearch( s, t=prod( j=1,L,prime(j)^v[L-j+1] )) & next; s=setunion(s,Set(t)); a=concat(a,t)); apply(x->print1(x","),vecsort(a))) }

cp's OEIS Frontend

A113198 Tetrahedron T(g, h, q) = number of rotes of weight g, height h, quench q.

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A178484 For n=1,2,... list all numbers not occurring earlier which can be written as a product of the first n primes raised to some nonnegative power less than n.

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A111794 Integers whose rote weight and rote height are equal, sorted by the equated value.

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A112480 Positive integers sorted by rote weight, rote wagage and rote height.

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A112481 Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.

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A112870 Positive integers sorted by rote height and primal code characteristic.

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A111799 Triangle T(h, w) = number of rotes of height h and wayage w.

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A113199 Positive integers sorted by rote weight, rote quench and rote height.

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A113200 Tetrahedron T(g, q, h) = number of rotes of weight g, quench q, height h.

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A178479 For n=0,1,2,... list all numbers not occurring earlier which can be written as product of the first n primes raised to some nonnegative power not exceeding n.

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