A063008 -id:A063008 - cp's OEIS frontend

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64, 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128, 9, 16, 21, 24, 25, 28, 36, 40, 45, 48, 60, 64, 72, 80, 81, 96, 108, 128, 144, 192, 256

Offset: 0

Amiram Eldar, Jun 16 2022

First differs from A074139 at the 8th row.
The n-th row begins with n+1, which corresponds to powers of primes, and ends with 2^n, which corresponds to squarefree numbers.
The n-th row contains the distinct values of the n-th row of A238963.

			Table begins:
  1;
  2;
  3, 4;
  4, 6, 8;
  5, 8, 9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64;
  8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128;
  ...
Numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Therefore the second row is {3, 4}.

Amiram Eldar, Table of n, a(n) for n = 0..18645 (rows 0..32, flattened)

Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).

row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten

row(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); Vec(m) } \\ Rémy Sigrist, Jun 17 2022

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 32, 12, 64, 24, 36, 128, 48, 72, 256, 96, 144, 30, 512, 192, 216, 288, 60, 1024, 384, 432, 576, 120, 2048, 768, 864, 180, 1152, 240, 1296, 4096, 1536, 1728, 360, 2304, 480, 2592, 8192, 3072, 3456, 720, 900, 4608, 960, 5184, 1080, 16384

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

The number of second signatures represented by the divisors of A181800(n) equals the number of prime signatures represented among the divisors of a(n). Cf. A212172, A212644.

A permutation of A025487.

			6 (whose prime factorization is 2*3) is the largest squarefree divisor of 144 (whose prime factorization is 2^4*3^2). Since 144 = A181800(10), and 144/6 = 24, a(10) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A003557, A007947, A085082, A181800, A212172, A212642, A212644.

Other permutations of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.

a(n) = A003557(A181800(n)).

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

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

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

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

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

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

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

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

cp's OEIS Frontend

A355026 Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

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A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

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

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A238970 The number of nodes at even level 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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Maple

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

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Examples

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Formula

Extensions

A238967 Maximal size of an antichain in canonical order.

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