cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 15, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 26, 1, 1, 2, 2, 1, 3, 1, 15, 2, 1, 1, 11, 1, 1, 1, 5, 1, 11, 1, 2, 1, 1, 1, 52, 1, 2, 2, 7, 1, 3, 1, 5, 3
Offset: 1

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Author

Gus Wiseman, Jul 22 2018

Keywords

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

Examples

			The a(36) = 7 maximal chains:
  (2*2*3*3) < (2*2*9) < (2*18) < (36)
  (2*2*3*3) < (2*2*9) < (4*9)  < (36)
  (2*2*3*3) < (2*3*6) < (2*18) < (36)
  (2*2*3*3) < (2*3*6) < (3*12) < (36)
  (2*2*3*3) < (2*3*6) < (6*6)  < (36)
  (2*2*3*3) < (3*3*4) < (3*12) < (36)
  (2*2*3*3) < (3*3*4) < (4*9)  < (36)
		

Crossrefs

Programs

  • PARI
    A064988(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); factorback(f); }; \\ From A064988
    memoA320105 = Map();
    A320105(n) = if(bigomega(n)<=2,1,if(mapisdefined(memoA320105,n), mapget(memoA320105,n), my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA320105,n,s); (s)));
    A317145(n) = A320105(A064988(n)); \\ Antti Karttunen, Oct 08 2018

Formula

a(prime^n) = A002846(n).
a(n) = A320105(A064988(n)). - Antti Karttunen, Oct 08 2018

Extensions

Data section extended to 105 terms by Antti Karttunen, Oct 08 2018

A317144 Number of refinement-ordered pairs of factorizations of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 14, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 26, 3, 3, 3, 31, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 56, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 55, 3, 12, 1, 9, 3, 12, 1, 82, 1, 3, 9, 9, 3, 12, 1, 56, 14
Offset: 1

Views

Author

Gus Wiseman, Jul 22 2018

Keywords

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
As this is a sequence computed from exponents in factorization of n, distinct values of a(n) in this sequence can be found by computing a(A025487(k)) for k >= 0. - David A. Corneth, Jul 30 2018

Examples

			The a(12) = 9 ordered pairs:
  (2*2*3) <= (12)
  (2*2*3) <= (2*6)
  (2*2*3) <= (3*4)
  (2*2*3) <= (2*2*3)
    (2*6) <= (12)
    (2*6) <= (2*6)
    (3*4) <= (12)
    (3*4) <= (3*4)
     (12) <= (12)
		

Crossrefs

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
    mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
    faccaps[fac_]:=Union[Sort/@Apply[Times,mps[fac],{2}]];
    Table[Sum[Length[faccaps[fac]],{fac,facs[n]}],{n,100}]

Formula

a(n) >= A001055(n) + floor(A000005(n) / 2) - 1. - David A. Corneth, Jul 30 2018

A317176 Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 18, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 49, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 74, 1, 4, 1, 3, 1, 4, 1, 78, 1, 1, 3, 3, 1, 4, 1, 49, 6, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Jul 23 2018

Keywords

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

Examples

			The a(24) = 11 chains:
  (2*2*2*3) < (24)
  (2*2*2*3) < (2*12)  < (24)
  (2*2*2*3) < (3*8)   < (24)
  (2*2*2*3) < (4*6)   < (24)
  (2*2*2*3) < (2*2*6) < (24)
  (2*2*2*3) < (2*3*4) < (24)
  (2*2*2*3) < (2*2*6) < (2*12) < (24)
  (2*2*2*3) < (2*2*6) < (4*6)  < (24)
  (2*2*2*3) < (2*3*4) < (2*12) < (24)
  (2*2*2*3) < (2*3*4) < (3*8)  < (24)
  (2*2*2*3) < (2*3*4) < (4*6)  < (24)
		

Crossrefs

Formula

a(prime^n) = A213427(n).

A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

Original entry on oeis.org

24, 32, 40, 48, 54, 56, 60, 64, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 243, 248, 250, 252, 256, 260, 264, 270
Offset: 1

Views

Author

Gus Wiseman, Jul 30 2018

Keywords

Comments

Includes 2^k for all k > 4.
Conjecture: Let S be the set of all numbers whose prime signature is either {1,3}, {5}, or {1,1,2}. Then the sequence consists of all multiples of elements of S. - David A. Corneth, Jul 31 2018.

Examples

			In the poset of factorizations of 24, the factorizations (2*2*6) and (2*3*4) have two least-upper bounds, namely (2*12) and (4*6), so this poset is not a lattice.
		

References

  • R. P Stanley, Enumerative Combinatorics Vol. 1, Sec. 3.3.

Crossrefs

A369713 a(n) is the sum over all multiplicative partitions k of n of the absolute value of the Möbius function evaluated at k,n in the poset of multiplicative partitions of n under refinement.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 11, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 11, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 5, 2, 1, 16, 2
Offset: 1

Views

Author

Tian Vlasic, Jan 29 2024

Keywords

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
For every natural number n, a(n) only depends on the prime signature of n.
a(n) is even if and only if n is a composite number.
Conjecture: There exists c such that a(n) <= n^c for all natural numbers n.

Examples

			The factorizations of 60 followed by their Moebius values are the following:
 (2*2*3*5) -> -3
 (2*2*15) ->  1
 (2*3*10) ->  2
 (2*5*6) ->  2
 (2*30) -> -1
 (3*4*5) ->  2
 (3*20) -> -1
 (4*15) -> -1
 (5*12) -> -1
 (6*10) -> -1
 (60) ->  1
Thus a(60)=16.
		

Crossrefs

Showing 1-5 of 5 results.