A096825 -id:A096825 - cp's OEIS frontend

A347042 Number of divisors d > 1 of n such that bigomega(d) divides bigomega(n), where bigomega = A001222.

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

Offset: 1

Gus Wiseman, Aug 17 2021

nonn

			The a(n) divisors for selected n:
  n = 1:   2:   4:   6:   24:  30:  36:  60:  96:  144: 210: 216: 240: 360:
      ---------------------------------------------------------------------
      {}   2    2    2    2    2    2    2    2    2    2    2    2    2
                4    3    3    3    3    3    3    3    3    3    3    3
                     6    4    5    4    4    4    4    5    4    4    4
                          6    30   6    5    6    6    6    6    5    5
                          24        9    6    8    8    7    8    6    6
                                    36   10   12   9    10   9    8    8
                                         15   96   12   14   12   10   9
                                         60        18   15   18   12   10
                                                   144  21   27   15   12
                                                        35   216  20   15
                                                        210       30   18
                                                                  240  20
                                                                       30
                                                                       45
                                                                       360

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of 1's are A000040.
The smallest of these divisors is A020639
The case of divisors with half bigomega is A345957 (rounded: A096825).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors, also called bigomega.
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors.
Cf. A026424, A033676, A033677, A335433, A335448, A347045, A347046.

Table[Length[Select[Rest[Divisors[n]],IntegerQ[PrimeOmega[n]/PrimeOmega[#]]&]],{n,100}]

a(n) = my(bn=bigomega(n)); sumdiv(n, d, if (d>1, !(bn % bigomega(d)))); \\ Michel Marcus, Aug 18 2021

from sympy import divisors, primeomega
def a(n):
    bigomegan = primeomega(n)
    return sum(bigomegan%primeomega(d) == 0 for d in divisors(n)[1:])
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021

from sympy import factorint, divisors
from sympy.utilities.iterables import multiset_combinations
def A347042(n):
    fs = factorint(n,multiple=True)
    return sum(len(list(multiset_combinations(fs,d))) for d in divisors(len(fs),generator=True)) # Chai Wah Wu, Aug 21 2021

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 07 2024

nonn

First differs from A096825 at a(210) = 4, A096825(210) = 6.
First differs from A343943 at a(210) = 4, A343943(210) = 6.
First differs from A345926 at a(90) = 4, A345926(90) = 3.

			For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
  (2*2*3*5): {{1,1,2,3}}
   (2*2*15): {{1,1,2},{1,1,3}}
   (2*3*10): {{1,1,2},{1,2,3}}
    (2*5*6): {{1,1,3},{1,2,3}}
    (3*4*5): {{1,2,3}}
     (2*30): {{1,1},{1,2},{1,3}}
     (3*20): {{1,2},{2,3}}
     (4*15): {{1,2},{1,3}}
     (5*12): {{1,3},{2,3}}
     (6*10): {{1,1},{1,2},{1,3},{2,3}}
       (60): {{1},{2},{3}}
So a(60) = 4.

Max Alekseyev, Table of n, a(n) for n = 1..10000

For all divisors (not just prime factors) we have A370816.
The version for partitions is A370809, for all divisors A370808.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]

A096826 Number of maximal-sized antichains in divisor lattice D(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 4, 3, 1, 2, 3, 2, 1, 1, 5, 2, 3, 2, 3, 1, 1, 2, 6, 3, 1, 4, 3, 2, 2, 2, 6, 1, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 3, 3, 1, 2, 10, 3, 3, 1, 3, 2, 6, 1, 6, 1, 1, 2, 1, 2, 1, 3, 7, 1, 2, 2, 3, 1, 2, 2, 4, 2, 1, 3, 3, 1, 2, 2, 10, 5, 1, 2, 1, 1, 1

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

nonn

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

			From _Gus Wiseman_, Aug 24 2018: (Start)
The a(120) = 6 antichains:
  {8,12,20,30}
  {8,12,15,20}
  {8,10,12,15}
  {6,8,15,20}
  {6,8,10,15}
  {4,6,10,15}
(End)

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A000005, A008480, A096825, A096827, A253249, A285572 A285573.

def A096826(n) :
    if n==1 : return 1
    R. = QQ[]; mults = [x[1] for x in factor(n)]
    maxsize = prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
    dlat = LatticePoset((divisors(n), attrcall("divides")))
    count = 0
    for ac in dlat.antichains_iterator() :
        if len(ac) == maxsize : count += 1
    return count
# Eric M. Schmidt, May 13 2013

More terms from Eric M. Schmidt, May 13 2013

A238946 Maximal level size of arcs in divisor lattice D(n).

Original entry on oeis.org

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

Offset: 1

Sung-Hyuk Cha, Mar 07 2014

nonn

A divisor d of n has level given by bigomega(d) and in-degree given by omega(d). The number of arcs on a level is the sum of the in-degrees of all divisors on the level. - Andrew Howroyd, Mar 28 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1000 (terms 1..200 from Sung-Hyuk Cha)
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. A001221 (omega), A001222 (bigomega), A062799, A096825, A238955, A238968.

a(n)={if(n==1, 0, my(v=vector(bigomega(n))); fordiv(n, d, if(d>1, v[bigomega(d)] += omega(d))); vecmax(v))} \\ Andrew Howroyd, Mar 28 2020

a(1) corrected by Andrew Howroyd, Mar 28 2020

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 19 2021

nonn

First differs from A096825 at a(525) = 3, A096825(525) = 4.
First differs from A345926 at a(90) = 4, A345926(90) = 3.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.

			The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.

The half-length submultisets are counted by A114921.
Including all multisets of prime factors gives A305611(n) + 1.
The strict rounded version appears to be counted by A342343.
The version for prime indices instead of prime factors is A345926.
A000005 counts divisors, which add up to A000203.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 and A299701 count positive subset-sums of partitions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A334968 counts subsequence-sums of standard compositions.
Cf. A008549, A032443, A083399, A096825, A344609, A345957.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A343943(n):
    fs = factorint(n)
    return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A238954 Maximal size of an antichain in graded colexicographic order of exponents.

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, 7, 8, 10, 14, 20, 1, 2, 3, 4, 4, 6, 7, 8, 8, 11, 13, 15, 18, 25, 35, 1, 2, 3, 4, 5, 4, 6, 8, 9, 10, 8, 12, 14, 16, 19, 16, 22, 26, 30, 36, 50, 70, 1, 2, 3, 4, 5, 4, 6, 8, 9, 9, 11, 12, 8, 12, 15, 17, 19, 22, 16, 23, 26, 30, 35, 31, 41, 48, 56, 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,  7, 8, 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. A096825 in graded colexicographic order.

Cf. A036035, A238967.

\\ 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)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Apr 25 2020

T(n,k) = A096825(A036035(n,k)).

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

cp's OEIS Frontend

A347042 Number of divisors d > 1 of n such that bigomega(d) divides bigomega(n), where bigomega = A001222.

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A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

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A096826 Number of maximal-sized antichains in divisor lattice D(n).

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A238946 Maximal level size of arcs in divisor lattice D(n).

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A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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A238954 Maximal size of an antichain in graded colexicographic order of exponents.

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A238967 Maximal size of an antichain in canonical order.

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