A299755 -id:A299755 - cp's OEIS frontend

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890

Offset: 0

Christian G. Bower, Dec 15 1999

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

			Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - _Philippe Deléham_, Jun 03 2015
From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........8                  12......../ \........24
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       10         15       30         20       40         60      120
  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840
  etc.
Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191 (Terms 0..1023 from T. D. Noe)
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"
Index entries for sequences that are permutations of the natural numbers

Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

up_to_e = 13; \\ Good for computing up to n = (2^13)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005
The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
From Antti Karttunen, Apr 12 & 17 2018: (Start)
a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
(End)

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

A299757 Weight of the strict integer partition with FDH number n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

Offset: 0

Alois P. Heinz, Sep 05 2014

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

			The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3;
   5,  6;
   7, 10;
  11, 14, 15;
  13, 21, 22, 30;
  17, 26, 33, 35, 42;
  19, 34, 39, 55, 66,  70;
  23, 38, 51, 65, 77,  78, 105, 110;
  29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
  ...
Corresponding triangle of strict integer partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (42) (51) (321)
        (7) (61) (52) (43) (421)
     (8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018

Alois P. Heinz, Rows n = 0..42, flattened

Column k=1 gives: A008578(n+1).
Last elements of rows give: A246868.
Row sums give A147655.
Row lengths are: A000009.
Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
      map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A334441 Maximum part of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 2, 3, 2, 1, 5, 3, 4, 2, 3, 2, 1, 6, 3, 4, 5, 2, 3, 4, 2, 3, 2, 1, 7, 4, 5, 6, 3, 3, 4, 5, 2, 3, 4, 2, 3, 2, 1, 8, 4, 5, 6, 7, 3, 4, 4, 5, 6, 2, 3, 3, 4, 5, 2, 3, 4, 2, 3, 2, 1, 9, 5, 6, 7, 8, 3, 4, 4, 5, 5, 6, 7, 3, 3, 4, 4, 5, 6, 2, 3, 3

Offset: 0

Gus Wiseman, May 06 2020

First differs from A049085 at a(8) = 2, A049085(8) = 3.

The parts of a partition are read in the usual (weakly decreasing) order. The version for reversed (weakly increasing) partitions is A049085.

			Triangle begins:
  0
  1
  2 1
  3 2 1
  4 2 3 2 1
  5 3 4 2 3 2 1
  6 3 4 5 2 3 4 2 3 2 1
  7 4 5 6 3 3 4 5 2 3 4 2 3 2 1
  8 4 5 6 7 3 4 4 5 6 2 3 3 4 5 2 3 4 2 3 2 1

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The length of the same partition is A036043.
Ignoring partition length (sum/lex) gives A036043 also.
The version for reversed partitions is A049085.
a(n) is the maximum element in row n of A334301.
The number of distinct parts in the same partition is A334440.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A001221, A103921, A124734, A185974, A296774, A299755, A334302, A334433, A334434, A334435, A334438.

Table[If[n==0,{0},Max/@Sort[IntegerPartitions[n]]],{n,0,10}]

A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3

Offset: 0

Gus Wiseman, May 05 2020

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.

			Triangle begins:
  0
  1
  1 1
  1 2 1
  1 1 2 2 1
  1 2 2 2 2 2 1
  1 1 2 2 1 3 2 2 2 2 1
  1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
  1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The number of not necessarily distinct parts is A036043.
The version for reversed partitions is A103921.
Ignoring length (sum/lex) gives A103921 (also).
a(n) is the number of distinct elements in row n of A334301.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]

a(n) = A001221(A334433(n)).

A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 6, 8, 5, 8, 9, 10, 6, 11, 12, 10, 7, 12, 6, 13, 8, 12, 15, 14, 8, 15, 9, 14, 10, 20, 18, 16, 11, 16, 12, 17, 10, 18, 21, 24, 12, 19, 18, 20, 13, 20, 24, 21, 12, 28, 15, 22, 14, 22, 24, 23, 15, 30, 27, 32, 14, 24, 30, 24, 20, 25

Offset: 1

Gus Wiseman, Jun 10 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = s_1 * ... * s_k.

Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Tree of x -> a(x) for n = 1...75

Cf. A003963, A050376, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A305830, A305831, A305832.

nn=100;
FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Times@@(FDfactor[n]/.FDrules),{n,nn}]

\\ here isfd is membership test for A050376.
isfd(n)={my(e=isprimepower(n)); e && e == 1<Andrew Howroyd, Aug 02 2018

A319247 Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 6, 4, 1, 3, 2, 7, 8, 4, 2, 5, 1, 9, 6, 1, 10, 3, 2, 1, 11, 5, 2, 7, 1, 4, 3, 12, 8, 1, 6, 2, 13, 4, 2, 1, 14, 9, 1, 15, 7, 2, 16, 5, 3, 8, 2, 10, 1, 17, 18, 11, 1, 6, 3, 5, 2, 1, 19, 9, 2, 4, 3, 1, 20, 21, 12, 1, 5, 4, 6, 2, 1, 22

Offset: 1

Gus Wiseman, Sep 15 2018

			The sequence of strict partitions begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (3,2,1), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Row lengths are A072047. Rows sums are A319246.

Cf. A000009, A000720, A001222, A005117, A056239, A265668, A296150, A299755.

Table[If[n==1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,Select[Range[100],SquareFreeQ]}]

T(n,k) = A000720(A265668(n,-k)).

A329631 Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 5, 6, 1, 4, 2, 3, 7, 8, 2, 4, 1, 5, 9, 1, 6, 10, 1, 2, 3, 11, 2, 5, 1, 7, 3, 4, 12, 1, 8, 2, 6, 13, 1, 2, 4, 14, 1, 9, 15, 2, 7, 16, 3, 5, 2, 8, 1, 10, 17, 18, 1, 11, 3, 6, 1, 2, 5, 19, 2, 9, 1, 3, 4, 20, 21, 1, 12, 4, 5, 1, 2, 6

Offset: 1

Gus Wiseman, Nov 18 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
   1: {}         33: {2,5}      66: {1,2,5}     97: {25}
   2: {1}        34: {1,7}      67: {19}       101: {26}
   3: {2}        35: {3,4}      69: {2,9}      102: {1,2,7}
   5: {3}        37: {12}       70: {1,3,4}    103: {27}
   6: {1,2}      38: {1,8}      71: {20}       105: {2,3,4}
   7: {4}        39: {2,6}      73: {21}       106: {1,16}
  10: {1,3}      41: {13}       74: {1,12}     107: {28}
  11: {5}        42: {1,2,4}    77: {4,5}      109: {29}
  13: {6}        43: {14}       78: {1,2,6}    110: {1,3,5}
  14: {1,4}      46: {1,9}      79: {22}       111: {2,12}
  15: {2,3}      47: {15}       82: {1,13}     113: {30}
  17: {7}        51: {2,7}      83: {23}       114: {1,2,8}
  19: {8}        53: {16}       85: {3,7}      115: {3,9}
  21: {2,4}      55: {3,5}      86: {1,14}     118: {1,17}
  22: {1,5}      57: {2,8}      87: {2,10}     119: {4,7}
  23: {9}        58: {1,10}     89: {24}       122: {1,18}
  26: {1,6}      59: {17}       91: {4,6}      123: {2,13}
  29: {10}       61: {18}       93: {2,11}     127: {31}
  30: {1,2,3}    62: {1,11}     94: {1,15}     129: {2,14}
  31: {11}       65: {3,6}      95: {3,8}      130: {1,3,6}

Row sums are A319246.
Row lengths are A072047.
Same as A319247 with rows reversed.
Composition of A000720 and A265668.
Looking at all numbers instead of just squarefree numbers gives A112798.
Cf. A000009, A005117, A048672, A056239, A299755, A302590.

Table[PrimePi/@First/@If[k==1,{},FactorInteger[k]],{k,Select[Range[30],SquareFreeQ]}]

A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 3, 2, 4, 1, 5, 3, 2, 1, 4, 2, 5, 1, 6, 4, 2, 1, 4, 3, 5, 2, 6, 1, 7, 4, 3, 1, 5, 2, 1, 5, 3, 6, 2, 7, 1, 8, 4, 3, 2, 5, 3, 1, 5, 4, 6, 2, 1, 6, 3, 7, 2, 8, 1, 9, 4, 3, 2, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 6, 4, 7, 2, 1, 7, 3, 8, 2, 9, 1, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (32)(41)(5)
  6: (321)(42)(51)(6)
  7: (421)(43)(52)(61)(7)
  8: (431)(521)(53)(62)(71)(8)
  9: (432)(531)(54)(621)(63)(72)(81)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of lex gives A118457.
The not necessarily strict version is A193073.
The version for reversed partitions is A246688.
The Heinz numbers of these partitions grouped by sum are A246867.
The ordered generalization is A339351.
Taking colex instead of lex gives A344087.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts reversed strict partitions by Heinz number.
A329631 sorts strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],lexsort],{n,0,8}]

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 5, 2, 3, 1, 4, 6, 2, 4, 1, 5, 1, 2, 3, 7, 3, 4, 2, 5, 1, 6, 1, 2, 4, 8, 3, 5, 2, 6, 1, 7, 1, 3, 4, 1, 2, 5, 9, 4, 5, 3, 6, 2, 7, 1, 8, 2, 3, 4, 1, 3, 5, 1, 2, 6, 10, 4, 6, 3, 7, 2, 8, 1, 9, 2, 3, 5, 1, 4, 5, 1, 3, 6, 1, 2, 7, 1, 2, 3, 4

Offset: 0

Gus Wiseman, May 12 2021

First differs from the revlex (instead of colex) version for partitions of 12.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(12)
  4: (4)(13)
  5: (5)(23)(14)
  6: (6)(24)(15)(123)
  7: (7)(34)(25)(16)(124)
  8: (8)(35)(26)(17)(134)(125)
  9: (9)(45)(36)(27)(18)(234)(135)(126)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724 plus one.
Taking lex instead of colex gives A026793 (non-reversed: A118457).
Triangle sums are A066189.
Reversing all partitions gives A344090.
The non-strict version is A344091.
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
Cf. A005117, A014466, A209862, A325683, A325859.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Table[Reverse/@Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}]

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