Sung-Hyuk Cha - cp's OEIS frontend

A238975 Number of perfect partitions in canonical order.

1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683, 64, 256, 544, 976, 768, 1888, 3408, 2316, 3172, 5740, 10404, 7880, 14300, 25988, 47293, 128, 576, 1376, 2496, 2208, 5536, 10096, 2568

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
   1;
   1;
   2,   3;
   4,   8,  13;
   8,  20,  26,  44,  75;
  16,  48,  76, 132, 176, 308,  541;
  32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
Sung-Hyuk Cha, Edgar G. DuCasse, Louis V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], (2014).
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See H(n) page 3.

Row sums are A035341.
Cf. A238962 in canonical order, A002033.
Cf. A063008, A074206.

g:= proc(n) option remember; (1+add(g(n/d),
      d=numtheory[divisors](n) minus {1, n}))
    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-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Apr 26 2020

(* b is A074206 *)
b[n_] := b[n] = If[n < 2, n, b /@ Most[Divisors[n]] // Total];
T[n_] := b /@ (Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]);
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jan 04 2021 *)

\\ here b(n) is A074206.
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
b(n)={if(!n, 0, my(sig=factor(n)[,2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k))))}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Aug 30 2020

T(n,k) = A074206(A063008(n,k)). - Andrew Howroyd, Apr 26 2020

Offset changed and terms a(42) and beyond from Andrew Howroyd, Apr 26 2020

A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665, 28, 70, 108, 165, 130, 240, 365, 268, 324, 492, 746, 594, 900, 1362, 2059, 36, 92, 147, 224, 186, 342, 519, 200, 410, 495, 750, 1135, 552, 836, 1008, 1524, 2302, 1215, 1836, 2772, 4182, 6305

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
   0;
   1;
   3,  5;
   6, 12, 19;
  10, 22, 27,  42, 65;
  15, 35, 48,  74, 90, 138, 211;
  21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665;
  ...

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. A238961 in canonical order.

Cf. A063008, A238952.

with(numtheory):
f:= n-> add(tau(d), d=divisors(n) minus {n}):
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-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

\\ here b(n) is A238952.
b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 26 2020

T(n,k) = A238952(A063008(n,k)). - Andrew Howroyd, Mar 26 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 26 2020

A238973 The number of arcs from odd to even level vertices in divisor lattice in canonical order.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 6, 2, 5, 6, 10, 16, 2, 6, 8, 14, 16, 26, 40, 3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96, 3, 9, 13, 22, 15, 29, 46, 32, 37, 58, 88, 67, 102, 152, 224, 4, 11, 16, 26, 19, 36, 56, 20, 41, 48, 74, 112, 52, 80, 93, 140, 208, 108, 162, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  0;
  1, 2;
  1, 3,  6;
  2, 5,  6, 10, 16;
  2, 6,  8, 14, 16, 26, 40;
  3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96;
  ...

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. A238960 in canonical order.

Cf. A063008, A238951, A238964, A238972.

From Andrew Howroyd, Mar 28 2020: (Start)
T(n,k) = A238951(A063008(n,k)).
T(n,k) = A238964(n,k) - A238972(n,k).
T(n,k) = floor(A238964(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 28 2020

A238972 The number of arcs from even to odd level vertices in divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96, 4, 10, 14, 22, 16, 30, 46, 32, 38, 58, 88, 68, 102, 152, 224, 4, 11, 16, 26, 19, 36, 56, 20, 41, 48, 74, 112, 52, 80, 93, 140, 208, 108, 162, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 4,  6;
  2, 5,  6, 10, 16;
  3, 7,  9, 14, 17, 26, 40;
  3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96;
  ...

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. A238959 in canonical order.

Cf. A063008, A238950, A238964, A238973.

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-> ceil((p-> add(nops(factorset(d)), d=divisors
    (p)))(mul(ithprime(i)^x[i], i=1..nops(x)))/2), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 28 2020

From Andrew Howroyd, Mar 28 2020: (Start)
T(n,k) = A238950(A063008(n,k)).
T(n,k) = A238964(n,k) - A238973(n,k).
T(n,k) = ceiling(A238964(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 28 2020

A238971 The number of nodes at odd level in divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 4, 8, 10, 14, 12, 18, 24, 12, 20, 22, 30, 40, 24, 32, 36, 48, 64, 40, 54, 72, 96, 128

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 3, 4;
  2, 4, 4,  6, 8;
  3, 5, 6,  8, 9, 12, 16;
  3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32;
  ...

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. A238958 in canonical order.

Cf. A056924, A063008, A238963, A238970.

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-> floor(numtheory[tau](mul(ithprime(i)
        ^x[i], i=1..nops(x)))/2), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 25 2020

b(n)={numdiv(n)\2}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

From Andrew Howroyd, Mar 25 2020: (Start)
T(n,k) = A056924(A063008(n,k)).
T(n,k) = A238963(n,k) - A238970(n,k).
T(n,k) = floor(A238963(n,k)/2). (End)

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

A238970 The number of nodes at even level in divisor lattice in canonical order.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 5, 8, 11, 14, 12, 18, 24, 13, 20, 23, 30, 40, 24, 32, 36, 48, 64, 41, 54, 72, 96, 128

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  1;
  1;
  2, 2;
  2, 3, 4;
  3, 4, 5,  6, 8;
  3, 5, 6,  8, 9, 12, 16;
  4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32;
  ...

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. A238957 in canonical order.
Leftmost column gives A008619.
Last terms of rows give A011782.
Cf. A038548, A063008, A238963, A238971.

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-> ceil(numtheory[tau](mul(ithprime(i)
        ^x[i], i=1..nops(x)))/2), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 25 2020

A063008row[n_] := Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n];
A038548[n_] := Ceiling[DivisorSigma[0, n]/2];
T[n_] := A038548 /@ A063008row[n];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Jan 30 2025 *)

\\ here b(n) is A038548.
b(n)={ceil(numdiv(n)/2)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

From Andrew Howroyd, Mar 25 2020: (Start)
T(n,k) = A038548(A063008(n,k)).
T(n,k) = A238963(n,k) - A238971(n,k).
T(n,k) = ceiling(A238963(n,k)/2). (End)

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

A238969 Degree of divisor lattice in divisor lattice in canonical order.

Original entry on oeis.org

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

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

Cf. A238956 in canonical order.

Cf. A063008, A238949.

C(sig)={sum(i=1, #sig, if(sig[i]>1, 2, 1))}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 26 2020

T(n,k) = A238949(A063008(n,k)). - Andrew Howroyd, Mar 26 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 26 2020

A238968 Maximal level size of arcs in divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 6, 1, 3, 4, 7, 12, 1, 3, 5, 8, 11, 18, 30, 1, 3, 5, 8, 6, 12, 19, 15, 24, 38, 60, 1, 3, 5, 8, 7, 13, 20, 16, 19, 30, 46, 37, 58, 90, 140, 1, 3, 5, 8, 7, 13, 20, 8, 17, 20, 31, 47, 23, 36, 43, 66, 100, 52, 80, 122, 185, 280

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  1, 3, 6;
  1, 3, 4, 7, 12;
  1, 3, 5, 8, 11, 18, 30;
  1, 3, 5, 8,  6, 12, 19, 15, 24, 38, 60;
  ...

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. A238955 in canonical order.

Cf. A063008, A238946.

\\ here b(n) is A238946.
b(n)={if(n==1, 0, my(v=vector(bigomega(n))); fordiv(n, d, if(d>1, v[bigomega(d)] += omega(d))); vecmax(v))}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 28 2020

T(n,k) = A238946(A063008(n,k)). - Andrew Howroyd, Mar 28 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 28 2020

A238967 Maximal size of an antichain in canonical order.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 7, 10, 1, 2, 3, 4, 4, 6, 8, 7, 10, 14, 20, 1, 2, 3, 4, 4, 6, 8, 7, 8, 11, 15, 13, 18, 25, 35, 1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 10, 14, 16, 22, 30, 19, 26, 36, 50, 70, 1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 9, 11, 15, 17, 23, 31, 12, 19, 26, 22, 30, 41, 56, 35, 48, 66, 91, 126

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 3, 4, 6;
  1, 2, 3, 4, 5, 7, 10;
  1, 2, 3, 4, 4, 6,  8, 7, 10, 14, 20;
  ...

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. A238954 in canonical order.

Cf. A063008, A096825.

with(numtheory):
f:= n-> (m-> add(`if`(bigomega(d)=m, 1, 0),
     d=divisors(n)))(iquo(bigomega(n), 2)):
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-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

\\ here b(n) is A096825.
b(n)={my(h=bigomega(n)\2); sumdiv(n, d, bigomega(d)==h)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

T(n,k) = A096825(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

User: Sung-Hyuk Cha

A238975 Number of perfect partitions in canonical order.

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A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.

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A238973 The number of arcs from odd to even level vertices in divisor lattice in canonical order.

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A238972 The number of arcs from even to odd level vertices in divisor lattice in canonical order.

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Maple

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A238971 The number of nodes at odd level in divisor lattice in canonical order.

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PARI

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A238970 The number of nodes at even level in divisor lattice in canonical order.

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Maple

Mathematica

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A238969 Degree of divisor lattice in divisor lattice in canonical order.

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