A106177 -id:A106177 - cp's OEIS frontend

A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 16 2019

Product of those distinct prime factors that occur exactly twice in the prime factorization of n, that is, whose exponent is 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000196, A062503, A337050.

Row 3 of array A106177, and the square roots of its row 9.

f[p_, e_] := If[e == 2, p, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 11 2023 *)

A329376(n) = { my(f = factor(n)); prod(i=1,#f~,f[i, 1]^(2 == f[i, 2])); };

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 + X + X^2/(-1 + 1/X)))[n], ", ")) \\ Vaclav Kotesovec, May 31 2024

Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.
a(n) <= A000196(n).
From Amiram Eldar, Feb 11 2023: (Start)
a(n) <= sqrt(n), with equality if and only if n is in A062503.
a(n) = 1 if and only if n is in A337050. (End)
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(2*s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
f'(1) = f(1) * Sum_{p prime} (9*p^2 - 12*p + 5) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 3.78385641685861932254178374972226733621783278751462026270346293...
and gamma is the Euler-Mascheroni constant A001620. (End)

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750

Offset: 1

Jon Awbrey, Jul 09 2005

nonn

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

Suggested by Franklin T. Adams-Watters

J. Awbrey, Riffs and Rotes
Rémy Sigrist, PARI program for A109299

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

PARI
```
\\ See Links section.
```

is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

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.

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.

A112872 First differences of A061396.

Original entry on oeis.org

0, 1, 4, 14, 53, 208, 843, 3494, 14769, 63378, 275480, 1210213, 5365307, 23973429, 107853409, 488137798, 2221048540, 10153825751, 46617145752, 214844862927, 993601026038, 4609693262811, 21448031058110, 100058764135997

Offset: 0

Jon Awbrey, Oct 24 2005, based on calculations by Vladeta Jovovic & David W. Wilson

nonn

J. Awbrey, Riffs and Rotes

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111801, A112846.

cp's OEIS Frontend

A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

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A109299 Primal codes of canonical finite permutations on positive integers.

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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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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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A112872 First differences of A061396.

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