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

%I A238966 #32 Sep 22 2023 08:43:53
%S A238966 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,
%T A238966 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,
%U A238966 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
%N A238966 The number of distinct primes in divisor lattice in canonical order.
%C A238966 After a(0) = 0, this appears to be the same as A128628. - _Gus Wiseman_, May 24 2020
%C A238966 Also the number of parts in the n-th integer partition in graded reverse-lexicographic order (A080577). - _Gus Wiseman_, May 24 2020
%H A238966 Andrew Howroyd, <a href="/A238966/b238966.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238966 S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], 2014.
%H A238966 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A238966 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A238966 T(n,k) = A001221(A063008(n,k)). - _Andrew Howroyd_, Mar 25 2020
%F A238966 a(n) = A001222(A129129(n)). - _Gus Wiseman_, May 24 2020
%e A238966 Triangle T(n,k) begins:
%e A238966   0;
%e A238966   1;
%e A238966   1, 2;
%e A238966   1, 2, 3;
%e A238966   1, 2, 2, 3, 4;
%e A238966   1, 2, 2, 3, 3, 4, 5;
%e A238966   1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6;
%e A238966   ...
%p A238966 o:= proc(n) option remember; nops(ifactors(n)[2]) end:
%p A238966 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238966     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238966 T:= n-> map(x-> o(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p A238966 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 26 2020
%t A238966 revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A238966 Table[Length/@Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* _Gus Wiseman_, May 24 2020 *)
%t A238966 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]]];
%t A238966 P[n_] := P[n] = Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
%t A238966 T[n_, k_] := PrimeNu[P[n][[k + 1]]];
%t A238966 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 *)
%o A238966 (PARI)
%o A238966 Row(n)={apply(s->#s, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238966 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020
%Y A238966 Row sums are A006128.
%Y A238966 Cf. A036043 in canonical order.
%Y A238966 Cf. A001221, A063008.
%Y A238966 Row lengths are A000041.
%Y A238966 The generalization to compositions is A000120.
%Y A238966 The sum of the partition is A036042.
%Y A238966 The lexicographic version (sum/lex) is A049085.
%Y A238966 Partition lengths of A080577.
%Y A238966 The partition has A115623 distinct elements.
%Y A238966 The Heinz number of the partition is A129129.
%Y A238966 The colexicographic version (sum/colex) is A193173.
%Y A238966 The maximum of the partition is A331581.
%Y A238966 Partitions in lexicographic order (sum/lex) are A193073.
%Y A238966 Partitions in colexicographic order (sum/colex) are A211992.
%Y A238966 Cf. A026792, A036036, A080576, A103921, A112798, A182715, A333486, A334302, A334435, A334436, A334442.
%K A238966 nonn,tabf
%O A238966 0,4
%A A238966 _Sung-Hyuk Cha_, Mar 07 2014
%E A238966 Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020