A063008 -id:A063008 - cp's OEIS frontend

A238966 The number of distinct primes in divisor lattice in canonical order.

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

After a(0) = 0, this appears to be the same as A128628. - Gus Wiseman, May 24 2020

Also the number of parts in the n-th integer partition in graded reverse-lexicographic order (A080577). - Gus Wiseman, May 24 2020

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 2, 3, 4;
  1, 2, 2, 3, 3, 4, 5;
  1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A006128.
Cf. A036043 in canonical order.
Cf. A001221, A063008.
Row lengths are A000041.
The generalization to compositions is A000120.
The sum of the partition is A036042.
The lexicographic version (sum/lex) is A049085.
Partition lengths of A080577.
The partition has A115623 distinct elements.
The Heinz number of the partition is A129129.
The colexicographic version (sum/colex) is A193173.
The maximum of the partition is A331581.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Cf. A026792, A036036, A080576, A103921, A112798, A182715, A333486, A334302, A334435, A334436, A334442.

o:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> o(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Table[Length/@Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 24 2020 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
P[n_] := P[n] = Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T[n_, k_] := PrimeNu[P[n][[k + 1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, Length[P[n]] - 1}] // Flatten (* Jean-François Alcover, Jan 03 2022, after Alois P. Heinz in A063008 *)

Row(n)={apply(s->#s, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

T(n,k) = A001221(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

a(n) = A001222(A129129(n)). - Gus Wiseman, May 24 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 25 2020

A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

Original entry on oeis.org

1, 2, 6, 4, 36, 30, 12, 8, 216, 180, 210, 900, 72, 60, 24, 16, 1296, 1080, 1260, 5400, 44100, 2310, 6300, 27000, 432, 360, 420, 1800, 144, 120, 48, 32, 7776, 6480, 7560, 32400, 264600, 13860, 37800, 162000, 9261000, 485100, 30030, 5336100, 1323000, 69300, 189000, 810000, 2592, 2160, 2520, 10800, 88200, 4620, 12600

Offset: 0

Antti Karttunen, Jan 16 2019

A101296(a(n)) gives a permutation of natural numbers.

			The sequence can be represented as a binary tree:
                                      1
                                      |
                   ...................2...................
                  6                                       4
       36......../ \........30                 12......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   216      180         210    900         72       60         24       16
etc.
Both children are multiples of their common parent, see A323503, A323504 and A323507.
The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
Sequence A227183 is based on the same algorithm.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000079 (right edge), A000400 (left edge, apart from 2), A005811, A046523, A101296, A227183, A322585, A322825, A323503, A323504, A323507.
Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.
Cf. A005940, A283477, A323505 for other similar trees.

{1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* Michael De Vlieger, Feb 05 2020 *)

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));

a(n) = A046523(a(n)) = A046523(A322825(n)).
A001221(a(n)) = A005811(n).
A001222(a(n)) = A227183(n).
A322585(a(n)) = 1.

A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1680, 1800, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 960, 1296, 1440

Offset: 0

Ray Chandler, Sep 04 2003

A025487 in a different order.

			1;
2;
4,6;
8,12,30;
16,24,36,60,210;
32,48,72,120,180,420,2310;
64,96,144,216,240,360,840,900,1260,4620,30030;
128,192,288,432,480,720,1080,1680,1800,2520,6300,9240,13860,60060,510510;

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A025487, A036035, A059901, A063008, A077569, A074140 (row sums), A328524.

b:= proc(n, i, l)
      `if`(n=0, [mul(ithprime(t)^l[t], t=1..nops(l))],
      `if`(i=1, b(0, 0, [l[], 1$n]), [b(n, i-1, l)[],
      `if`(i>n, [], b(n-i, i, [l[], i]))[]]))
    end:
T:= n-> sort(b(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 13 2012

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Join[{Product[Prime[t]^l[[t]], {t, 1, Length[l]}]}], If[i == 1, b[0, 0, Join[l, Table[1, {n}]]], Join[b[n, i - 1, l], If[i > n, {}, b[n - i, i, Append[l, i]]]]]];
T[n_] := Sort[b[n, n, {}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)

A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 08 2020

The first partition ranked by A080577 is (); there is no zeroth partition.

			The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.
  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (2,1,1,1)    (6,1)          (8)
  (3)        (1,1,1,1,1)  (5,2)          (7,1)
  (2,1)      (6)          (5,1,1)        (6,2)
  (1,1,1)    (5,1)        (4,3)          (6,1,1)
  (4)        (4,2)        (4,2,1)        (5,3)
  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
Triangle begins:
  0
  1
  2 1
  3 2 1
  4 3 2 2 1
  5 4 3 3 2 2 1
  6 5 4 4 3 3 3 2 2 2 1
  7 6 5 5 4 4 4 3 3 3 3 2 2 2 1
  8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
The version for compositions is A065120 or A333766.
Reverse-lexicographically ordered partitions are A080577.
Distinct parts of these partitions are counted by A115623.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Lengths of these partitions are A238966.
Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Prepend[First/@Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,8}],0]

a(n) = A061395(A129129(n - 1)).

A074140 Sum of least integers of prime signatures over all partitions of n.

Original entry on oeis.org

1, 2, 10, 50, 346, 3182, 38770, 609290, 11226106, 250148582, 7057182250, 216512001950, 7903965900226, 321552174623162, 13779150603234010, 644574260638821590, 33968684108427733426, 1994885097404292104942, 121496572792097514728530, 8114030083731371137603190

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Old name was: Sum of terms in n-th group in A036035.

a(n) is also the sum of terms in n-th row of A063008, A087443 or A227955.

			a(6) = 64+96+144+216+240+360+900+840+1260+4620+30030 = 38770.

Peter Luschny and Alois P. Heinz, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Cf. A036035, A063008, A074139, A074141, A025487, A087443, A227955, A332626.

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, ithprime(j)^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, If[i<1, 0, b[n, i-1, j]+If[i>n, 0, Prime[j]^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

def A074140(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        m = mul(P[i]^pi for i, pi in enumerate(p))
        L.append(m)
    return add(L)
[A074140(n) for n in (0..20)]  # Peter Luschny, Aug 02 2013

More terms from Alford Arnold, Sep 10 2002
a(10)-a(12) from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Sep 30 2004
a(12) corrected by Peter Luschny, Aug 03 2013
New name from Alois P. Heinz, Aug 03 2013

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

A330661 T(n,k) is the index within the partitions of n in canonical ordering of the k-th partition whose parts differ pairwise by at most one.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 5, 8, 9, 10, 11, 1, 5, 9, 12, 13, 14, 15, 1, 8, 13, 18, 19, 20, 21, 22, 1, 8, 19, 22, 26, 27, 28, 29, 30, 1, 13, 22, 30, 37, 38, 39, 40, 41, 42, 1, 13, 30, 41, 46, 51, 52, 53, 54, 55, 56, 1, 20, 44, 59, 62, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Peter Dolland, Dec 23 2019

For each length k in [1..n] there is exactly one such partition [p_1,...,p_k], with p_i = a+1 for i=1..j and p_i = a for i=j+1..k, where a = floor(n/k) and j = n - k * a.
If k | n, then all parts p_i are equal. A027750 lists the indices of these partitions in this triangle.
Canonical ordering is also known as graded reverse lexicographic ordering, see A080577 or link below.

			Partitions of 8 in canonical ordering begin: 8, 71, 62, 611, 53, 521, 5111, 44, 431, 422, 4211, 41111, 332, ... . The partitions whose parts differ pairwise by at most one in this list are 8, 44, 332, ... at indices 1, 8, 13, ... and this gives row 8 of this triangle.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  3,  4,  5;
  1,  3,  5,  6,  7;
  1,  5,  8,  9, 10, 11;
  1,  5,  9, 12, 13, 14, 15;
  1,  8, 13, 18, 19, 20, 21, 22;
  1,  8, 19, 22, 26, 27, 28, 29, 30;
  1, 13, 22, 30, 37, 38, 39, 40, 41, 42;
  ...

Alois P. Heinz, Rows n = 1..200, flattened
OEIS Wiki, Orderings of partitions (a comparison).

Cf. A000041, A063008, A027750, A080577, A216053, A238639, A238640.

T(2n,n) gives A332706.

b:= proc(l) option remember; (n-> `if`(n=0, 1,
      b(subsop(1=[][], l))+g(n, l[1]-1)))(add(j, j=l))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
T:= proc(n, k) option remember; 1 + g(n$2)-
      b((q-> [q+1$r, q$k-r])(iquo(n, k, 'r')))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 19 2020

b[l_List] := b[l] = Function[n, If[n == 0, 1, b[ReplacePart[l, 1 -> Nothing]] + g[n, l[[1]] - 1]]][Total[l]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
T[n_, k_] := T[n, k] = Module[{q, r}, {q, r} = QuotientRemainder[n, k]; 1 + g[n, n] - b[Join[Table[q + 1, {r}], Table[q, {k - r}]]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

balP(p) = p[1]-p[#p]<=1
Row(n)={v=vecsort([Vecrev(p) | p<-partitions(n)], , 4);select(i->balP(v[i]),[1..#v])}
{ for(n=1, 10, print(Row(n))) }

T(n,1) = 1.
T(n,n) = A000041(n).
T(n,k) = A000041(n) - (n - k) for k = ceiling(n/2)..n.
T(2n,2) = T(2n+1,2) = A216053(n). - Alois P. Heinz, Jan 28 2020

A328773 Irregular triangle read by rows: T(n,k) is the number of colored digraphs on n nodes with color scheme given by the partitions of n in canonical ordering.

Original entry on oeis.org

1, 1, 3, 4, 16, 36, 64, 218, 752, 1104, 2112, 4096, 9608, 45960, 90416, 178944, 266496, 528384, 1048576, 1540944, 9133760, 22692704, 45277312, 30194176, 90196736, 180011008, 135032832, 269500416, 537919488, 1073741824

Offset: 0

Peter Dolland, Oct 27 2019

Colors are not interchangeable. Adjacent nodes may have the same color.
A partition [b_1, ..., b_m] with b_i > 0 and Sum_{i=1..m} b_i = n corresponds to a color scheme on n nodes having m colors. To find out which digraphs are equivalent with respect to a color scheme, consider the automorphism group on the partition. This group is the m-fold product of the symmetric groups on the b_i nodes, and therefore contains Product_{i=1..m} b_i! elements (i.e. the permutations).
Calculate the number of equivalence classes by determining the cycle index of the group induced by the automorphism group in the set of the edges [(i,j)|i,j in [1..n]; i != j] and set, with Pólya, the variable values to 2.
The left column of the triangle gives the number of unlabeled digraphs, while the right flank of the triangle gives the number of labeled digraphs.
Canonical ordering is also known as graded reverse lexicographic ordering, see A080577, A063008, or link below. Partitions here have the property b_i >= b_j for i < j.

			The sequence begins:
      1;
      1;
      3,       4;
     16,      36,       64;
    218,     752,     1104,     2112,     4096;
   9608,   45960,    90416,   178944,   266496,   528384,   1048576;
   ...
For n = 3, the three partitions of n are [3], [2, 1] and [1, 1, 1]. T(n,1) = 16 gives the number of colored digraphs with all nodes having the same color; T(n, 2) = 36 gives the number of colored digraphs with two nodes having the first color and one node having the second color; T(n, 3) gives the number of colored digraphs with each node having its own color.
For n = 5, k = 4 the required partition is [3,1,1]. T(5,4) = 178944 is then the number of colored digraphs with 5 nodes, where 3 nodes have the first color and the other two nodes each has its own color.

N. G. de Bruijn, Pólyas Abzähl-Theorie: Muster für Graphen und chemische Verbindungen, Selecta Mathematica III, Springer-Verlag (1971), 1-55.

Peter Dolland, Table of n, a(n) for n = 0..138 (rows 0..10)
OEIS Wiki, Orderings of partitions (a comparison).

Cf. A000041 equals the row length, A080577 lists the partitions in the used order, A063008 instantiates the index sequences encoding the partitions. A000273 and A053763 represent the flanks of the triangle.

\\ here C(p) computes sequence value for given partition.
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i]-1)}
C(p)={((i,v)->if(i>#p, 2^edges(v), my(s=0); forpart(q=p[i], s+=permcount(q)*self()(i+1, concat(v,Vec(q)))); s/p[i]!))(1, [])}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], ,4))}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Nov 02 2019

T(n, 1) = A000273(n).
T(n, A000041(n)) = A053763(n) = 2^(n^2 - n).
T(n, A000041(n)-1) = 2^(n^2 - 3*n - 1) * (2^(2*n) + 8) for n > 1.

A062515 Leading least prime signatures, ordered by forming the product of primorials greater than 2 with multiplicities given by the canonical sequence of partitions.

Original entry on oeis.org

1, 6, 30, 36, 210, 180, 216, 2310, 1260, 900, 1080, 1296, 30030, 13860, 6300, 7560, 5400, 6480, 7776, 510510, 180180, 69300, 83160, 44100, 37800, 45360, 27000, 32400, 38880, 46656, 9699690, 3063060, 900900, 1081080, 485100, 415800, 498960, 264600, 189000

Offset: 0

Alford Arnold, Jul 10 2001

From Jack W Grahl, Jul 06 2018: (Start)
The least prime signatures (A025487) are the smallest numbers with a given 'prime signature'. For example, 24 = 2^3 * 3 is the smallest number consisting of the cube of a prime multiplied by a prime. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 >= k2 >= k3 >= ...
These can also be defined as all products of primorials. Here the primorials (A002110) are the products of the first n primes. So 24 = 2 * 2 * 6.
The leading least prime signatures (A056153) are the least prime signatures k such that k/2 is not a least prime signature. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 = k2 >= k3 >= ... (note the first operator is equality). A056153 lists these in increasing order.
They can also be defined as all products of primorials 6 or greater. This sequence lists the leading least prime signatures in an ordering derived from this definition. The canonical sequence of partitions maps to this sequence under a mapping which sends 1 -> 6, 2 -> 30, 3 -> 210, etc., and then forms the product of these terms. Thus the first few partitions are [], [1], [2], [1,1], [3], [2,1] and so the first terms of this sequence are 1, 6, 30, 6 * 6 = 36, 210, 30 * 6 = 180.
The previous description described this sequence as the 'leading least prime signatures ordered as in A063008'. This was in error. A063008 gives a different ordering of A025487, also based on the canonical sequence of partitions, but the definition is different from this sequence and the terms do not appear in the same order (with the transposition of 216 and 2310 being the first discrepancy). (End)

			Values in A025487 can be generated via powers of two as follows:
1
2
4,6
8,12
16,24,30
32,48,60,36
64,96,120,72
128,192,240,144,210,180,216
a(3) = 36 because we can write [1,1] and associate this exponent vector with 6*6

Jack W Grahl, Table of n, a(n) for n = 0..1000

Cf. A002110, A025487, A056153 and A063008.

import Data.List(inits)
primes :: [Integer]
primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
    where
  divides a b = a `mod` b == 0
primorials :: [Integer]
primorials = map product $ inits primes
partitions :: [[Integer]]
partitions = concat $ map (partitions_of_n) [0..]
partitions_of_n :: Integer -> [[Integer]]
partitions_of_n n = partitions_at_most n n
partitions_at_most :: Integer -> Integer -> [[Integer]]
partitions_at_most _ 0 = [[]]
partitions_at_most 0 _ = []
partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
a062515 :: [Integer]
a062515 = map primorial_signature partitions
    where
  primorial_signature p = product $ map ((drop 1 primorials) !!) (map fromIntegral p)
-- Jack W Grahl, Jul 06 2018

Clarified and extended by Jack W Grahl, Jul 06 2018

A238964 Size of divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 12, 4, 10, 12, 20, 32, 5, 13, 17, 28, 33, 52, 80, 6, 16, 22, 36, 24, 46, 72, 54, 84, 128, 192, 7, 19, 27, 44, 31, 59, 92, 64, 75, 116, 176, 135, 204, 304, 448, 8, 22, 32, 52, 38, 72, 112, 40, 82, 96, 148, 224, 104, 160, 186, 280, 416, 216, 324, 480, 704, 1024

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  2,  4;
  3,  7, 12;
  4, 10, 12, 20, 32;
  5, 13, 17, 28, 33, 52, 80;
  6, 16, 22, 36, 24, 46, 72, 54, 84, 128, 192;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Cf. A238953 in canonical order.

Cf. A062799, A063008.

with(numtheory):
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> (p-> add(nops(factorset(d)), d=divisors
    (p)))(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 24 2020

A062799[n_] := PrimeNu[Divisors[n]] // Total;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
A063008[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T[n_] := A062799 /@ A063008[n];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Apr 16 2025, after Alois P. Heinz in A063008 *)

\\ here b(n) is A062799.
b(n)={sumdiv(n, d, omega(d))}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{concat(vector(9, n, Row(n-1)))} \\ Andrew Howroyd, Mar 24 2020

T(n,k) = A062799(A063008(n,k)). - Andrew Howroyd, Mar 24 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 24 2020

cp's OEIS Frontend

A238966 The number of distinct primes in divisor lattice in canonical order.

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A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

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A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

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A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

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A074140 Sum of least integers of prime signatures over all partitions of n.

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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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A330661 T(n,k) is the index within the partitions of n in canonical ordering of the k-th partition whose parts differ pairwise by at most one.

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