A122172 -id:A122172 - cp's OEIS frontend

A122402 A sequence of rectangular tables with row sums A122401 related to A002865 and A122172.

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

Offset: 0

Alford Arnold, Sep 02 2006

Sequence A002865 gives the number of rows in each rectangular table. A122402 is a subsequence of A122172.

			There are four cyclic partitions of 6: 6, 42, 33 and 222; therefore the corresponding table is
1 1 1 1 1 1 1
1 2 3 3 3 2 1
1 2 3 4 3 2 1
1 3 6 7 6 3 1
The tables begin:
1
0
1 1 1
1 1 1 1
1 1 1 1 1
1 2 3 2 1
1 1 1 1 1 1
1 2 3 3 2 1
...

Cf. A002865 A122172 A122401.

A122453 Triangular array related to A122402 as A079025 relates to A122172; sums A079139 values associated with cyclic partitions.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 2, 3, 4, 4, 3, 2, 4, 8, 13, 15, 13, 8, 4, 4, 8, 13, 16, 16, 13, 8, 4, 7, 17, 32, 45, 51, 45, 32, 17, 7, 8, 20, 38, 56, 67, 67, 56, 38, 20, 8, 12, 34, 72, 117, 156, 171, 156, 117, 72, 34, 12, 14, 41, 88, 147, 203, 237, 237, 203, 147, 88, 41

Offset: 0

Alford Arnold, Sep 07 2006

			The table begins:
1
0 0
1 1 1
1 1 1 1
2 3 4 3 2
2 3 4 4 3 2
4 8 13 15 13 8 4
4 8 13 16 16 13 8 4
7 17 32 45 51 45 32 17 7
8 20 38 56 67 67 56 38 20 8
12 34 72 117 156 171 156 117 72 34 12
14 41 88 147 203 237 237 203 147 88 41 14

Cf. A002865 A074139 A079025 A122172 A122402.

A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 25, 28, 30, 27, 40, 36, 48, 64, 17, 26, 33, 35, 44, 42, 50, 45, 56, 60, 54, 80, 72, 96, 128, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256, 23, 38, 51, 65, 77, 68, 78, 110, 98, 99, 105, 125, 104, 132, 140, 126, 150, 135, 176, 168, 200, 180, 162, 224, 240, 216, 320, 288, 384, 512, 29, 46, 57, 85, 91, 121, 76, 102, 130, 154, 117, 165, 147, 175, 136, 156, 220, 196, 198, 210, 250, 189, 225, 208, 264, 280, 252, 300, 270, 243, 352, 336, 400, 360, 324, 448, 480, 432, 640, 576, 768, 1024

Offset: 0

Wolfdieter Lang, Feb 10 2011

First differs from A334438 (shifted left once) at a(75) = 98, A334438(76) = 99. - Gus Wiseman, May 20 2020
This mapping of the set of all partitions of N >= 0 to {1, 2, 3, ...} (set of natural numbers) is one to one (bijective). The empty partition for N = 0 maps to 1.
A129129 seems to be analogous, except that the partition ordering A080577 is used. This ordering, however, does not care about the number of parts: e.g., 1^2,4 = 4,1^2 comes before 3^2, so a(23)=28 and a(22)=25 are interchanged.
Also Heinz numbers of all reversed integer partitions (finite weakly increasing sequences of positive integers), sorted first by sum, then by length, and finally lexicographically, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The version for non-reversed partitions is A334433. - Gus Wiseman, May 20 2020

			a(22) = 25 = prime(3)^2 because the 22nd partition in A-St order is the 2-part partition (3,3) of N = 6, because A026905(5) = 18 < 22 <= A026905(6) = 29.
a(23) = 28 = prime(1)^2*prime(4) corresponds to the partition 1+1+4 = 4+1+1 with three parts, also of N = 6.
From _Gus Wiseman_, May 20 2020: (Start)
Triangle begins:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  25  28  30  27  40  36  48  64
  17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
As a triangle of reversed partitions we have:
                             0
                            (1)
                          (2)(11)
                        (3)(12)(111)
                   (4)(13)(22)(112)(1111)
             (5)(14)(23)(113)(122)(1112)(11111)
  (6)(15)(24)(33)(114)(123)(222)(1113)(1122)(11112)(111111)
(End)

M. F. Hasler, Table of n, a(n) for n = 0..9295 (up to partitions of 25), Jan 07 2024
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order.

Row lengths are A000041.
The constructive version is A036036.
Also Heinz numbers of the partitions in A036037.
The generalization to compositions is A124734.
The version for non-reversed partitions is A334433.
The non-reversed length-insensitive version is A334434.
The opposite version (sum/length/revlex) is A334435.
Ignoring length gives A334437.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Partitions in colexicographic order are A211992.
Graded Heinz numbers are A215366.
Cf. A026791, A036043, A056239, A080577, A228531, A296150, A334301, A334302, A334436, A334438, A334439.

Join@@Table[Times@@Prime/@#&/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 21 2020 *)

A185974_row(n)=[vecprod([prime(i)|i<-p])|p<-partitions(n)] \\ below a helper function:
index_of_partition(n)={for(r=0, oo, my(c = numbpart(r)); n >= c || return([r,n+1]); n -= c)}
/* A185974(n,k), 1 <= k <= A000041(n), gives the k-th partition of n >= 0; if k is omitted, A185974(n) return the term of index n of the flattened sequence a(n >= 0).
  This function is used in other sequences (such as A122172) which need to access the n-th partition as listed in A-S order. */
A185974(n, k=index_of_partition(n))=A185974_row(iferr(k[1], E, k=[k,k]; n))[k[2]] \\ (End)

a(n) = Product_{j=1..N(n)} p(j)^e(j), with p(j):=A000040(j) (j-th prime), and the exponent e(j) >= 0 of the part j in the n-th partition written in Abramowitz-Stegun (A-St) order, indicated in A036036. Note that j^0 is not 1 but has to be omitted in the partition. N(n) is the index (argument) of the smallest A026905-number greater than or equal to n (the index of the A026905-ceiling of n).
From Gus Wiseman, May 21 2020: (Start)
A001221(a(n)) = A103921(n).
A001222(a(n)) = A036043(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A049085(n).
(End)

Examples edited by M. F. Hasler, Jan 07 2024

A122401 Subsequence of A074139 omitting values derived from partitions with a part of size 1.

Original entry on oeis.org

1, 3, 4, 5, 9, 6, 12, 7, 15, 16, 27, 8, 18, 20, 36, 9, 21, 24, 25, 45, 48, 81, 10, 24, 28, 30, 54, 60, 64, 108, 11, 27, 32, 35, 63, 36, 72, 75, 80, 135, 144, 243, 12, 30, 36, 40, 72, 42, 84, 90, 96, 162, 100, 180, 192, 324, 13, 33, 40, 45, 81, 48, 96, 49, 105, 112, 189, 108, 120, 216, 125, 225, 240, 405, 256, 432, 729, 14, 36, 44, 50, 90, 54

Offset: 0

Alford Arnold, Sep 01 2006

When viewed as a table, row sums are given by sequence A079274.

Corresponds to members of A036035 which are also powerful numbers (A001694).

			The two cyclic partitions of five are 5 and 3+2 yielding (5+1)=6 and (3+1)*(2+1) = 4*3 = 12
The array begins
1
(empty)
3
4
5 9
6 12
7 15 16 27
8 18 20 36

Cf. A122172 A079274 A122402, A316532.

A122401_row := proc(n)
    local e, a,L;
    L := [] ;
    for e in ListTools[Reverse](partition(n)) do
        if member(1,e) then
            ;
        else
            a := 1;
            for p in e do
                a := a*(p+1) ;
            end do:
            L := [op(L),a] ;
        end if;
    end do:
    L ;
end proc:
seq(A122401_row(i), i=0..15); # R. J. Mathar, Aug 28 2018

Extended by R. J. Mathar, Aug 28 2018

cp's OEIS Frontend

A122402 A sequence of rectangular tables with row sums A122401 related to A002865 and A122172.

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A122453 Triangular array related to A122402 as A079025 relates to A122172; sums A079139 values associated with cyclic partitions.

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A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.

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A122401 Subsequence of A074139 omitting values derived from partitions with a part of size 1.

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A122403 Table read by rows: rearrangement of A122401 to illustrate linear column sequences.

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