A114717 -id:A114717 - cp's OEIS frontend

A119840 Positions where new distinct values appear in A114717.

1, 6, 12, 24, 30, 36, 60, 72, 96, 120, 144, 180, 192, 210, 216, 240, 288, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 900, 960, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2160, 2304, 2310, 2520, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4320, 4608, 4620

Offset: 1

Antti Karttunen, May 26 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..69

Subset of A025487. Cf. A119500, A119841.

b[s_] := b[s] = If[Length[s] < 2, 1, Sum[If[Length[Select[s, Mod[#, x] == 0 &]] == 1, b[Complement[s, {x}]], 0], {x, s}]]; a[n_] := a[n] = Module[{l, m}, l = Sort[FactorInteger[n], #1[[2]] > #2[[2]] &]; m = Product[Prime[i ]^l[[ i]][[2]], {i, 1, Length[l]}]; b[Divisors[m][[2 ;; -2]] ]]; A119840 = Reap[For[k = 1, k < 5000, k++, ak = a[k]; If[FreeQ[Array[a, k - 1], ak], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(19)-a(53) from Alois P. Heinz, Aug 06 2012

A119499 Records in A114717.

Original entry on oeis.org

1, 2, 5, 14, 48, 2452, 183958, 4877756, 17454844, 20071150430, 409158464142, 129586764260850, 269333638458151764, 1868569007661198289216, 326772188939088357313806, 48024472200935599107697461965204, 11653191042139941668276738496190656

Offset: 1

Antti Karttunen, May 26 2006

nonn

Term a(10) was taken from A114716(4) (= A114715(3,4) = A114715(4,3)).

Alois P. Heinz, Table of n, a(n) for n = 1..20

a(n) = A114717(A119500(n)). Subset of A119841.

b[s_] := b[s] = If[Length[s] < 2, 1, Sum[If[Length[Select[s, Mod[#, x] == 0 &]] == 1, b[Complement[s, {x}]], 0], {x, s}]]; a[n_] := Module[{l, m}, l = Sort[FactorInteger[n], #1[[2]] > #2[[2]] &]; m = Product[Prime[i]^l[[i]][[2]], {i, 1, Length[l]}]; b[Divisors[m] // Rest // Most]]; A119499 = Reap[For[record = 0; k = 1, k < 5000, k++, If[a[k] > record, record = a[k]; Print[k, " ", record]; Sow[record]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(11)-a(17) from Alois P. Heinz, Aug 06 2012

A119500 Numbers n such that A114717(n) sets a new record.

Original entry on oeis.org

1, 6, 12, 24, 30, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 4620, 5040, 7560, 9240

Offset: 1

Antti Karttunen, May 26 2006

nonn

This sequence gives yet another list of "increasingly compound" numbers.

Cf. A119499, A018894. Subset of A119840.

b[s_] := b[s] = If[Length[s] < 2, 1, Sum[If[Length[Select[s, Mod[#, x] == 0 &]] == 1, b[Complement[s, {x}]], 0], {x, s}]]; a[n_] := a[n] = Module[{l, m}, l = Sort[FactorInteger[n], #1[[2]] > #2[[2]] &]; m = Product[Prime[i ]^l[[i]][[2]], {i, 1, Length[l]}]; b[Divisors[m] // Rest // Most]]; A119500 = Reap[For[record = 0; k = 1, k < 10^4, k++, ak = a[k]; If[ak > record, record = ak; Print[k, " ", ak]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(11)-a(17) from Alois P. Heinz, Aug 06 2012

a(18)-a(20) from Alois P. Heinz, Feb 25 2016

A119841 Distinct values in A114717 in order of appearance.

Original entry on oeis.org

1, 2, 5, 14, 48, 42, 2452, 462, 132, 183958, 6006, 4877756, 429, 1680384, 24024, 17454844, 87516, 20071150430, 1430, 409158464142, 1662804, 1941406508, 1385670, 129586764260850, 4862, 269333638458151764, 140229804, 6405442434150, 242201554680, 6708527580006468

Offset: 1

Antti Karttunen, May 26 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..69

a(n) = A114717(A119840(n)). Cf. A119499.

b[s_] := b[s] = If[Length[s] < 2, 1, Sum[If[Length[Select[s, Mod[#, x] == 0 &]] == 1, b[Complement[s, {x}]], 0], {x, s}]]; a[n_] := a[n] = Module[{l, m}, l = Sort[FactorInteger[n], #1[[2]] > #2[[2]] &]; m = Product[Prime[ i]^l[[i]][[2]], {i, 1, Length[l]}]; b[Divisors[m] // Rest // Most]]; A119841 = Reap[For[k = 1, k < 2000, k++, If[FreeQ[Array[a, k - 1], a[k]], Print[k, " ", a[k]]; Sow[a[k]]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(19)-a(30) from Alois P. Heinz, Aug 06 2012

A119842 Number of alternating linear extensions of the divisor lattice of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 6, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 1, 0

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

For prime powers there is only one solution. For integers with prime signature p1^2 * p2 there's exactly one solution, for p1^4 * p2 there are two and in general for p1^(2k) * p2 there are A000108(k) solutions. - Mitch Harris, Apr 27 2006

			In other words, the number of ways to arrange the divisors of n in such a way that no divisor has any of its own divisors following it AND the divisors d_i, d_j, d_k, etc. are arranged so that values bigomega(d_i) (cf. A001222), bigomega(d_j), bigomega(d_k) are alternatively even and odd. E.g., a(12)=1, as of the five arrangements shown in A114717, here the only one allowed is 1,2,4,3,6,12, with A001222(1)=0, A001222(2)=1, A001222(4)=2, A001222(3)=1, A001222(6)=2, A001222(12)=3. a(36) = 2, as there are two solutions for 36: 1,2,4,3,6,12,9,18,36 and 1,3,9,2,6,18,4,12,36.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
T. Y. Chow, H. Eriksson, C. K. Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol. 11(2), 2004.
T. Y. Chow, H. Eriksson, C. K. Fan, Chess Tableaux and Chess Problems, slides for MIT Combinatorics Seminar, 20 October 2004.
Index entries for sequences computed from exponents in factorization of n

a(n) <= A114717(n). Cf. A119844, A119846, A119847, A119849.

with(numtheory):
b:= proc(s, t) option remember; `if`(nops(s)<1, 1, add(
      `if`(irem(bigomega(x), 2)=1-t and nops(select(y->
      irem(y, x)=0, s))=1, b(s minus {x}, 1-t), 0), x=s))
    end:
a:= proc(n) option remember; local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(divisors(m) minus {1, m}, irem(bigomega(m), 2))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 26 2016

b[s_, t_] := b[s, t] = If[Length[s] < 1, 1, Sum[If[Mod[PrimeOmega[x], 2] == 1-t && Length[Select[s, Mod[#, x] == 0&]] == 1, b[s ~Complement~ {x}, 1-t ], 0], {x, s}]]; a[n_] := a[n] = Module[{l, m}, l = Sort[FactorInteger[n ], #1[[2]] > #2[[2]]&]; m = Product[Prime[i]^l[[i]][[2]], {i, 1, Length[ l]}]; b[Divisors[m][[2 ;; -2]], Mod[PrimeOmega[m], 2]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A114714 Number of linear extensions of a 2 X 2 X n lattice.

Original entry on oeis.org

1, 2, 48, 2452, 183958, 17454844, 1941406508, 242201554680, 32959299267334, 4801233680739724, 738810565910888784, 118929992674840615128, 19880920716640427983476, 3431624482227380273056728, 608880419873586515669564728, 110654016191338341346670548240

Offset: 0

Mitch Harris, Dec 27 2005

The additional terms were found using dynamic programming to count the maximal chains in the distributive lattice of order-preserving functions from the chain of length n to J(P), where J is the lattice of downsets of the poset P = 2x2. - Nick Krempel, Jul 08 2010

Stanley, R., Enumerative Combinatorics, Vol. 2, Prop. 7.10.3 and Vol. 1, Sec 3.5, Chains in Distributive Lattices.

Alois P. Heinz, Table of n, a(n) for n = 0..180 (first 41 terms from Nick Krempel)

Cf. A114717.

b:= proc(x, y, u, w) option remember;
      `if`(x=0 and y=0 and u=0 and w=0, 1, `if`(x>y and x>u,
       b(x-1, y, u, w), 0)+ `if`(y>w, b(x, y-1, u, w), 0)+
      `if`(u>w, b(x, y, u-1, w), 0)+ `if`(w>0, b(x, y, u, w-1), 0))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 27 2012

b[x_, y_, u_, w_] := b[x, y, u, w] = If[x == 0 && y == 0 && u == 0 && w == 0, 1, If[x>y && x>u, b[x-1, y, u, w], 0] + If[y>w, b[x, y-1, u, w], 0] + If[u>w, b[x, y, u-1, w], 0] + If[w>0, b[x, y, u, w-1], 0]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)

More terms from Nick Krempel, Jul 08 2010

A114715 Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 5, 48, 5, 14, 2452, 2452, 14, 42, 183958, 4877756, 183958, 42, 132, 17454844, 20071150430, 20071150430, 17454844, 132, 429, 1941406508, 129586764260850, 6708527580006468, 129586764260850, 1941406508, 429

Offset: 1

Mitch Harris, Dec 27 2005

			Square array A(n,m) begins:
   1,      2,           5,               14, ...
   2,     48,        2452,           183958, ...
   5,   2452,     4877756,      20071150430, ...
  14, 183958, 20071150430, 6708527580006468, ...

Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.

Alois P. Heinz, Antidiagonals n = 1..12, flattened

Cf. A000108, A114714, A114716, A114717.

Main diagonal gives A370257.

b := proc(l) option remember; local n; n:= nops(l);
      `if`({seq(l[i][], i=1..n)}={0}, 1, add(`if`(l[i][1]>l[i][2] and
       l[i][1]>l[i+1][1], b(subsop(i=[l[i][1]-1, l[i][2]], l)), 0),
       i=1..n-1)+ add(`if`(l[i][2]>l[i+1][2], b(subsop(i=[l[i][1],
       l[i][2]-1], l)), 0), i=1..n-1)+ `if`(l[n][1]>l[n][2],
       b(subsop(n=[l[n][1]-1, l[n][2]], l)), 0)+ `if`(l[n][2]>0,
       b(subsop(n=[l[n][1], l[n][2]-1], l)), 0))
     end:
A:= (n, m)-> `if`(m>=n, b([[m$2]$n]), b([[n$2]$m])):
seq(seq(A(n, d+1-n), n=1..d), d=1..8);  # Alois P. Heinz, Jun 29 2012

b[l_List] := b[l] = With[{n = Length[l]}, If[Union[Table[l[[i]], {i, 1, n}] // Flatten] == {0}, 1, Sum[If[l[[i, 1]] > l[[i, 2]] && l[[i, 1]] > l[[i+1, 1]], b[ReplacePart[l, i -> {l[[i, 1]]-1, l[[i, 2]]}]], 0], {i, 1, n-1}] + Sum[If[l[[i, 2]] > l[[i+1, 2]], b[ReplacePart[l, i -> {l[[i, 1]], l[[i, 2]]-1}]], 0], {i, 1, n-1}] + If[l[[n, 1]] > l[[n, 2]], b[ReplacePart[l, n -> {l[[n, 1]]-1, l[[n, 2]]} ]], 0] + If[l[[n, 2]] > 0, b[ReplacePart[l, n -> {l[[n, 1]], l[[n, 2]]-1}]], 0]]] ; A[n_, m_] := If[m >= n, b[Array[{m, m}&, n]], b[Array[{n, n}&, m]]]; Table[ Table[A[n, d+1-n], {n, 1, d}], {d, 1, 8}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

A(n,1) = A(1,n) = A000108(n).
A(n,2) = A(2,n) = A114714(n).
A(n,3) = A(3,n) = A114716(n).

Edited by Alois P. Heinz, Jun 29 2012

A114716 Number of linear extensions of a 3 X 2 X n lattice.

Original entry on oeis.org

1, 5, 2452, 4877756, 20071150430, 129586764260850, 1138355914222027660, 12513844842339741519760, 163186564770917385358723138, 2434438822161210367337209525489, 40488679486377745566571570522228550, 736610570835499716578578298705683198672

Offset: 0

Mitch Harris, Dec 27 2005; corrected May 25 2006

Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.

Alois P. Heinz, Table of n, a(n) for n = 0..45

Cf. A114717.

b:= proc(u, v, w, x, y, z) option remember;
      `if`({u, v, w, x, y, z}={0}, 1,
      `if`(u>v and u>x, b(u-1, v, w, x, y, z), 0)+
      `if`(v>w and v>y, b(u, v-1, w, x, y, z), 0)+
      `if`(w>z, b(u, v, w-1, x, y, z), 0)+
      `if`(x>y, b(u, v, w, x-1, y, z), 0)+
      `if`(y>z, b(u, v, w, x, y-1, z), 0)+
      `if`(z>0, b(u, v, w, x, y, z-1), 0))
    end:
a:= n-> b(n$6):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 26 2012

b[u_, v_, w_, x_, y_, z_] := b[u, v, w, x, y, z] =
If[Union[{u, v, w, x, y, z}] == {0}, 1,
If[u>v && u>x, b[u-1, v, w, x, y, z], 0] +
If[v>w && v>y, b[u, v-1, w, x, y, z], 0] +
If[w>z, b[u, v, w-1, x, y, z], 0] +
If[x>y, b[u, v, w, x-1, y, z], 0] +
If[y>z, b[u, v, w, x, y-1, z], 0] +
If[z>0, b[u, v, w, x, y, z-1], 0]];
a[n_] := b[n, n, n, n, n, n]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)

a(6)-a(11) from Alois P. Heinz, Apr 26 2012

A163820 Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 36, 1, 2, 2, 24, 1, 36, 1, 36, 2, 2, 1, 1440, 2, 2, 6, 36, 1, 348, 1, 120, 2, 2, 2, 10560, 1, 2, 2, 1440, 1, 348, 1, 36, 36, 2, 1, 100800, 2, 36, 2, 36, 1, 1440, 2, 1440, 2, 2, 1, 2218560, 1, 2, 36, 720, 2, 348, 1, 36, 2, 348, 1, 9737280, 1, 2, 36, 36, 2, 348, 1, 100800, 24, 2, 1, 2218560, 2, 2, 2, 1440, 1, 2218560, 2, 36, 2, 2, 2, 10886400, 1, 36, 36, 10560

Offset: 1

Leroy Quet, Aug 04 2009

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(60) = a(90) since 60 = 2^2 * 3 * 5 and 90 = 2 * 3^2 * 5 both have prime signature (2,1,1). - Antti Karttunen, Oct 22 2017

As a consequence of the comment above, a(n) = a(A046523(n)). - David A. Corneth, Oct 22 2017

			The divisors of 12 that are > 1 are 2,3,4,6,12. In the permutations that are counted, 3 cannot be next to 2 or 4. However, a permutation that is among those counted is 6,2,4,12,3. The GCDs of adjacent pairs in this permutation are gcd(6,2)=2, gcd(2,4)=2, gcd(4,12)=4, gcd(12,3)=3. Note that all of these GCDs are > 1.

David A. Corneth, Table of n, a(n) for n = 1..359 (first 119 terms from Antti Karttunen)
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. also A046523, A114717, A119842, A122977, A122978, A293900, A293902.

Array[Count[Permutations@ Rest@ Divisors[#], ?(NoneTrue[Partition[#, 2, 1], CoprimeQ @@ # &] &)] - Boole[# == 1] &, 59] (* _Michael De Vlieger, Nov 04 2017 *)

a(p) = 1 for all primes p. a(p*q) = 2 for all pairs of (not necessarily distinct) primes p and q.
From Antti Karttunen, Oct 22 2017: (Start)
a(p^n) = A000142(n), for all primes p.
a(n) = A293900(n)*A293902(n).
(End)

Definition corrected by Leroy Quet, Aug 15 2009

Edited and extended by Max Alekseyev, Jun 13 2011

A293900 Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime and no divisor d may occur later than any divisor e if e < d and A007947(e) = A007947(d). That is, any subset of divisors sharing the same squarefree part occur always in ascending order inside the permutation.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 9, 1, 2, 2, 1, 1, 9, 1, 9, 2, 2, 1, 40, 1, 2, 1, 9, 1, 348, 1, 1, 2, 2, 2, 110, 1, 2, 2, 40, 1, 348, 1, 9, 9, 2, 1, 175, 1, 9, 2, 9, 1, 40, 2, 40, 2, 2, 1, 138660, 1, 2, 9, 1, 2, 348, 1, 9, 2, 348, 1, 1127, 1, 2, 9, 9, 2, 348, 1, 175, 1, 2, 1, 138660, 2, 2, 2, 40, 1, 138660, 2, 9, 2, 2, 2, 756, 1, 9, 9, 110

Offset: 1

Antti Karttunen, Oct 22 2017

nonn

This is a more restricted variant of A163820, inspired by David A. Corneth's suggestion (personal e-mail) for optimizing its computation.

			The proper divisors of 12 are 2, 3, 4, 6, 12. a(12) = 9 because we can find nine permutations of them such that consecutive elements d and e are not coprime (that is, gcd(d,e) > 1) and where no divisor d is ever followed by divisor e such that A007947(d) = A007947(e) and e < d. These nine allowed permutations are (note that 2 must become before 4 and 6 must become before 12):
  [2, 4, 6, 3, 12],
  [2, 4, 6, 12, 3],
  [2, 6, 3, 12, 4],
  [2, 6, 4, 12, 3],
  [3, 6, 2, 4, 12],
  [3, 6, 2, 12, 4],
  [3, 6, 12, 2, 4],
  [6, 2, 4, 12, 3],
  [6, 3, 12, 2, 4].

Antti Karttunen, Table of n, a(n) for n = 1..179 (based on b-file of A163820 provided by David A. Corneth)
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A000961 (positions of 0 and 1's), A163820, A293902.

Cf. also A114717, A119842.

Iff n = p^k for some prime p and k >= 1 [that is, n is a term of A000961 > 1], then a(n) = 1.

a(n) = A163820(n)/A293902(n).

cp's OEIS Frontend

A119840 Positions where new distinct values appear in A114717.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A119499 Records in A114717.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A119500 Numbers n such that A114717(n) sets a new record.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A119841 Distinct values in A114717 in order of appearance.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A119842 Number of alternating linear extensions of the divisor lattice of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A114714 Number of linear extensions of a 2 X 2 X n lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A114715 Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A114716 Number of linear extensions of a 3 X 2 X n lattice.

Views

Author

Keywords

References

Links