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

0,4

Comments

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

Links

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

Formula

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

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

Example

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;
  ...

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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.

Sequence in context: A036043 A333486 A128628 * A353510 A334230 A275723

Adjacent sequences: A238963 A238964 A238965 * A238967 A238968 A238969

Keyword

nonn,tabf

Author

Sung-Hyuk Cha, Mar 07 2014

Extensions

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

Status

approved