A119842 -id:A119842 - cp's OEIS frontend

A119847 Positions where A119842 is zero.

6, 10, 14, 15, 21, 22, 24, 26, 30, 33, 34, 35, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

In addition to the terms of A119899 includes also terms with prime signature p*q*r, e.g. 30 (= 2*3*5). Anything else?

Alois P. Heinz, Table of n, a(n) for n = 1..6677

Complement: A119848. Superset of A119899. Cf. A119850.

Cf. A229153.

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}]]; c[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]]]; A119847 = Position[ Table[c[n], {n, 1, 200}], 0] // Flatten (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A119848 Positions where A119842 is not zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..3323

Complement: A119847. Cf. A119850.

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}]]; c[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]]]; A119848 = Position[ Table[c[n], {n, 1, 200}], ?Positive] // Flatten (* _Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A119845 Positions where new distinct values appear in A119842.

Original entry on oeis.org

1, 6, 36, 72, 120, 144, 180, 192, 288, 420, 432, 480, 576, 720, 900, 1080, 1152, 1296, 1680, 1728, 1800, 1920, 2304, 2520, 2592, 2880, 3072, 3600, 4320, 4608, 5184, 6300, 6480, 6720, 6912, 7200, 7680, 9216, 9240

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

a(16) is greater than 960.

Subset of A025487. Cf. A119846, A119843.

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]]]; Join[{1}, Reap[For[ k = 2, k < 10^4, k++, If[FreeQ[Array[a, k - 1], a[k]], Print[k, " ", a[k]]; Sow[k]]]][[2, 1]]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(16)-a(39) from Alois P. Heinz, Feb 26 2016

A119846 Distinct values in A119842, in order of appearance.

Original entry on oeis.org

1, 0, 2, 6, 14, 22, 252, 5, 92, 17526, 324, 2916, 422, 9278038, 411126, 23694308, 2074, 8716, 1186593143406, 47570, 11395487424, 918166, 10754, 789632365568454684, 343234, 1164751580868, 42, 917581820575340, 1914786011328142, 58202, 17423496

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..39

a(n) = A119842(A119845(n)).

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]]]; A119846 = Reap[For[k = 1, k < 6000, 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(16)-a(31) from Alois P. Heinz, Feb 26 2016

A119850 Positions where A119842 is greater than one.

Original entry on oeis.org

36, 48, 72, 80, 100, 108, 112, 120, 144, 162, 168, 176, 180, 192, 196, 200, 208, 225, 252, 264, 270, 272, 280, 288, 300, 304, 312, 320, 324, 368, 378, 392, 396, 400, 405, 408, 420, 432, 440, 441, 448, 450, 456, 464, 468, 480, 484, 496, 500, 520, 552, 567

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1281

Subset of A119848.

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_] := 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]]]; A119842 = Array[a, 1000]; A119850 = Position[A119842, n_ /; n > 1] // Flatten (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A119843 A119842(n) sets a new record.

Original entry on oeis.org

1, 36, 72, 120, 144, 180, 420, 720, 1080, 1680, 2520, 6300, 9240

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

a(9) is greater than 960.

Subset of A119845. Cf. A119844.

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]]]; Reap[For[record = 0; k = 1, k < 10^4, k++, If[a[k] > record, record = a[k]; Print[k, " ", a[k]]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(9)-a(13) from Alois P. Heinz, Feb 26 2016

A119844 Records in A119842.

Original entry on oeis.org

1, 2, 6, 14, 22, 252, 17526, 9278038, 23694308, 1186593143406, 789632365568454684, 3199548374111958458274, 849440501364681630324594977448

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

a(n) = A119842(A119843(n)).

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]]]; Reap[For[record = 0; k = 1, k < 10^4, k++, If[a[k] > record, record = a[k]; Print[k, " ", a[k]]; Sow[a[k]]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

a(9)-a(13) from Alois P. Heinz, Feb 26 2016

A119849 Nonzero values of A119842, in order of appearance.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 14, 1, 1, 1, 1, 1, 2, 1, 252, 1, 1, 1, 5, 1, 2

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..3323

a(n) = A119842(A119848(n)).

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]]]; Reap[For[ k = 1, k < 200, k++, If[a[k] > 0, Print[k, " ", a[k]]; Sow[a[k]]]]][[2, 1]] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A114717 Number of linear extensions of the divisor lattice of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 5, 1, 5, 2, 2, 1, 14, 1, 2, 1, 5, 1, 48, 1, 1, 2, 2, 2, 42, 1, 2, 2, 14, 1, 48, 1, 5, 5, 2, 1, 42, 1, 5, 2, 5, 1, 14, 2, 14, 2, 2, 1, 2452, 1, 2, 5, 1, 2, 48, 1, 5, 2, 48, 1, 462, 1, 2, 5, 5, 2, 48, 1, 42, 1, 2, 1, 2452, 2, 2, 2, 14, 1, 2452, 2

Offset: 1

Mitch Harris and Antti Karttunen, Dec 27 2005

nonn

Notice that only the powers of the primes determine a(n), so a(12) = a(75) = 5.
For prime powers, the lattice is a chain, so there is 1 linear extension.
a(p^1*q^n) = A000108(n+1), the Catalan numbers.
Alternatively, 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. E.g., for 12, the following five arrangements are possible: 1,2,3,4,6,12; 1,2,3,6,4,12; 1,2,4,3,6,12; 1,3,2,4,6,12 and 1,3,2,6,4,12. But 1,2,6,4,3,12 is not possible because 3 divides 6 but follows it. Thus a(12)=5. - Antti Karttunen, Jan 11 2006
For n = p1^r1 * p2^r2, the lattice is a grid (r1+1)*(r2+1), whose linear extensions are counted by ((r1+1)*(r2+1))!/Product_{k=0..r2} (r1+1+k)!/k!. Cf. A060854.

R. Stanley, 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 = 1..10000
Graham Brightwell and Peter Winkler, Counting linear extensions, Order 8 (1991), no. 3, 225-242.
Gary Pruesse and Frank Ruskey, Generating linear extensions fast, SIAM J. Comput. 23 (1994), no. 2, 373-386.
Index entries for sequences computed from exponents in factorization of n

Cf. A060854, A114714, A114715, A114716, A119842.

with(numtheory):
b:= proc(s) option remember;
      `if`(nops(s)<2, 1, add(`if`(nops(select(y->
       irem(y, x)=0, s))=1, b(s minus {x}), 0), x=s))
    end:
a:= proc(n) 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})
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2012

b[s_List] := 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]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

A179926 Number of permutations of the divisors of n of the form d_1=n, d_2, d_3, ..., d_tau(n) such that d_(i+1)/d_i is a prime or 1/prime for all i.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 18, 1, 1, 2, 2, 2, 8, 1, 2, 2, 4, 1, 18, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 106, 1, 2, 3, 1, 2, 18, 1, 3, 2, 18, 1, 17, 1, 2, 3, 3, 2, 18, 1, 5, 1, 2, 1, 106, 2, 2, 2, 4, 1, 106, 2, 3, 2, 2, 2, 6, 1, 3, 3, 8, 1, 18, 1, 4, 18, 2, 1, 17, 1, 18, 2, 5, 1, 18, 2, 3, 3, 2, 2, 572

Offset: 1

Vladimir Shevelev, Aug 02 2010

nonn

In view of formulas given below, there are many common first terms with A001221. Note that, for n >= 1, a(n) is positive; it is function of exponents of prime power factorization of n only; moreover, it is invariant with respect to permutations of them.
An equivalent multiset formulation of the problem: for a given finite multiset A, we should, beginning with A, to get all submultisets of A, if, by every step, we remove or join 1 element. How many ways are there to do this?
Via Seqfan Discussion List (Aug 03 2010), Alois P. Heinz proved that every subsequence of the form a(p), a(p*q), a(p*q*r), ..., where p, q, r, ... are distinct primes, coincides with A003043. - Vladimir Shevelev, Aug 09 2010
The parity (odd or even) of bigomega(d_i) in a permutation of divisors of n alternates. - David A. Corneth, Nov 25 2017
Equivalently, the number of Hamiltonian paths in a graph with vertices corresponding to the divisors of n and edges connecting divisors that differ by a prime with the path starting on the vertex associated with 1. - Andrew Howroyd, Oct 26 2019

			a(12)=3:
[12, 6, 3, 1, 2, 4]
[12, 4, 2, 6, 3, 1]
[12, 4, 2, 1, 3, 6]
a(45)=3:
[45, 15, 5, 1, 3, 9]
[45, 9, 3, 15, 5, 1]
[45, 9, 3, 1, 5, 15]

Alois P. Heinz, Table of n, a(n) for n = 1..1259 (first 719 terms from David A. Corneth)
David A. Corneth, The permutations of divisors for a(60) ordered by their last element.
V. Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 2011-2014.
V. Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001221, A180026, A003043, A003042. - Vladimir Shevelev, Aug 09 2010
See A173675 for another version.
Cf. A119842, A295785.

q:= (i, j)-> is(i/j, integer) and isprime(i/j):
b:= proc(s, l) option remember; `if`(s={}, 1, add(
     `if`(q(l, j) or q(j, l), b(s minus{j}, j), 0), j=s))
    end:
a:= n-> (s-> b(s minus {n}, n))(numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2017

q[i_, j_] := PrimeQ[i/j];
b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[q[l, j] || q[j, l], b[s  ~Complement~ {j}, j], 0], {j, s}]];
a[n_] := Function[s, b[s ~Complement~ {n}, n]][Divisors[n]];
Array[a, 120] (* Jean-François Alcover, Dec 13 2017, after Alois P. Heinz *)

a(n) = {my(f = factor(n), l = List(), chain = List()); res = 0; forvec(x = vector(#f~, i, [0, f[i, 2]]), listput(l, x)); listput(chain, l[#l]); listpop(l, #l); iterate(chain, l); res}
iterate(c, l) = {if(#l == 1, if(vecsum(abs(c[#c] - l[1])) == 1, res++), my(cc, cl);
for(i = 1, #l, if(vecsum(abs(c[#c] - l[i])) == 1, cc = c; cl = l; listput(cc, l[i]); listpop(cl, i); iterate(cc, cl))))}
first(n) = {my(res = vector(n), m = Map()); res[1] = 1; for(i = 2, n, cn = a046523(i); if(cn == i, mapput(m, i, a(i))); res[i] = mapget(m, cn)); res}
a046523(n)=my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f,(p=nextprime(p+1))^f[i]) \\ (a046523 from Charles R Greathouse IV), David A. Corneth, Nov 24 2017

a(p^k)=1, a(p^k*q)=k+1, a(p^2*q^2)=8, a(p^2*q^3)=17, a(pqr)=18, a(p^2*q*r)=106, a(p^3*q*r)=572, etc. (here p,q,r are distinct primes, k >= 0).

Corrected by D. S. McNeil and Alois P. Heinz and extended by Alois P. Heinz from a(46) via the Seqfan Discussion List (Aug 02 2010)

cp's OEIS Frontend

A119847 Positions where A119842 is zero.

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A119848 Positions where A119842 is not zero.

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A119845 Positions where new distinct values appear in A119842.

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A119846 Distinct values in A119842, in order of appearance.

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A119850 Positions where A119842 is greater than one.

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A119843 A119842(n) sets a new record.

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A119844 Records in A119842.

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A119849 Nonzero values of A119842, in order of appearance.

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A114717 Number of linear extensions of the divisor lattice of n.

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