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 1-10 of 11 results. Next

A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

Original entry on oeis.org

36, 60, 72, 84, 90, 100, 108, 120, 126, 132, 140, 144, 150, 156, 168, 180, 196, 198, 200, 204, 210, 216, 220, 225, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 315, 324, 330, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 390
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2018

Keywords

Comments

Positions of nonzero terms in A322437 or A322438.
Mats Granvik has conjectured that these are all the positive integers k such that sigma_0(k) - 2 > (bigomega(k) - 1) * omega(k), where sigma_0 = A000005, omega = A001221, and bigomega = A001222. - Gus Wiseman, Nov 12 2019
Numbers with more semiprime divisors than prime divisors. - Wesley Ivan Hurt, Jun 10 2021

Examples

			An example of such a pair for 36 is (4*9)|(6*6).

Links

Crossrefs

Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442.
The following are additional cross-references relating to Granvik's conjecture.
bigomega(n) * omega(n) is A113901(n).
(bigomega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - bigomega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[Subsets[facs[#],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]
  • PARI
    factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
has_at_least_one_ndfpair(z) = { for(i=1,#z,for(j=i+1,#z,if(is_ndf_pair(z[i],z[j]),return(1)))); (0); };
isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408
Offset: 1

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
            (111)       (11111)  (222)   (3211)     (3311)    (333)
                                 (3111)  (1111111)  (221111)  (4221)
                                                              (222111)
                                                              (111111111)

Crossrefs

For median instead of mean we have A362558.
The complement is counted by A362560.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A322442 Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.

Original entry on oeis.org

1, 1, 4, 25, 195, 1894, 22159, 303769, 4790858, 85715595, 1720097275, 38355019080, 942872934661, 25383601383937, 744118939661444, 23635548141900445, 809893084668253151, 29822472337116844174, 1175990509568611058299, 49504723853840395163221, 2218388253903492656783562
Offset: 0

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(3) = 25 pairs of set partitions (these are actually all pairs of set partitions of {1,2,3}):
  (1)(2)(3)|(1)(2)(3)
  (1)(2)(3)|(1)(23)
  (1)(2)(3)|(12)(3)
  (1)(2)(3)|(13)(2)
  (1)(2)(3)|(123)
    (1)(23)|(1)(2)(3)
    (1)(23)|(1)(23)
    (1)(23)|(12)(3)
    (1)(23)|(13)(2)
    (1)(23)|(123)
    (12)(3)|(1)(2)(3)
    (12)(3)|(1)(23)
    (12)(3)|(12)(3)
    (12)(3)|(13)(2)
    (12)(3)|(123)
    (13)(2)|(1)(2)(3)
    (13)(2)|(1)(23)
    (13)(2)|(12)(3)
    (13)(2)|(13)(2)
    (13)(2)|(123)
      (123)|(1)(2)(3)
      (123)|(1)(23)
      (123)|(12)(3)
      (123)|(13)(2)
      (123)|(123)
Non-isomorphic representatives of the pairs of set partitions of {1,2,3,4} for which the condition fails:
    (12)(34)|(13)(24)
    (12)(34)|(1)(3)(24)
  (1)(2)(34)|(13)(24)

Links

Crossrefs

Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A322435, A322437, A322439, A322440, A322441.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
costabQ[s_,t_]:=And@@Cases[s,x_:>Select[t,SubsetQ[x,#]||SubsetQ[#,x]&]!={}];
Table[Length[Select[Tuples[sps[Range[n]],2],And[costabQ@@#,costabQ@@Reverse[#]]&]],{n,5}]
  • PARI
    F(x)={my(bell=(exp(y*(exp(x) - 1))  )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( exp( 2*exp(exp(x) - 1) - exp(x) - 1) * F(x) ))} \\ Andrew Howroyd, Jan 19 2024

Formula

E.g.f.: exp(exp(x)-1) * (2*B(x) - 1) where B(x) is the e.g.f. of A319884. - Andrew Howroyd, Jan 19 2024

Extensions

a(8) onwards from Andrew Howroyd, Jan 19 2024

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 5, 0, 3, 0, 3, 1, 1, 0, 7, 1, 1, 2, 3, 0, 4, 0, 7, 1, 1, 1, 15, 0, 1, 1, 7, 0, 4, 0, 3, 3, 1, 0, 16, 1, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 18, 0, 1, 3, 16, 1, 4, 0, 3, 1, 4, 0, 32, 0, 1, 3, 3, 1, 4, 0, 16, 5, 1, 0, 18, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(36) = 15 pairs of factorizations:
  (2*2*3*3)|(4*9)
  (2*2*3*3)|(6*6)
  (2*2*3*3)|(36)
    (2*2*9)|(6*6)
    (2*2*9)|(36)
    (2*3*6)|(4*9)
    (2*3*6)|(36)
     (2*18)|(36)
    (3*3*4)|(6*6)
    (3*3*4)|(36)
     (3*12)|(36)
      (4*9)|(6*6)
      (4*9)|(36)
      (6*6)|(4*9)
      (6*6)|(36)

Crossrefs

Cf. A000688, A001055, A050336, A265947, A294068, A305149, A305150, A305193, A317144, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Tuples[facs[n],2],!Or@@Divisible@@@Tuples[#]&]],{n,100}]

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

Original entry on oeis.org

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098
Offset: 1

Views

Author

Gus Wiseman, Apr 28 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

Crossrefs

For median instead of mean we have A322439 aerated, complement A362558.
The complement is counted by A362559.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 8, 1, 3, 3, 11, 1, 8, 1, 8, 3, 3, 1, 18, 3, 3, 5, 8, 1, 12, 1, 15, 3, 3, 3, 31, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 39, 3, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 42, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 39, 11
Offset: 1

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(12) = 8 pairs of factorizations:
  (2*2*3)|(2*2*3)
  (2*2*3)|(2*6)
  (2*2*3)|(3*4)
  (2*2*3)|(12)
    (2*6)|(12)
    (3*4)|(3*4)
    (3*4)|(12)
     (12)|(12)

Crossrefs

Cf. A000688, A001055, A050336, A265947, A281113, A294068, A303386, A305149, A305150, A305193, A317144, A322435, A322439, A322440, A322441, A322442.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Tuples[facs[n],2],!Or@@divpropQ@@@Tuples[#]&]],{n,100}]

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

Original entry on oeis.org

1, 0, 1, 2, 5, 7, 16, 20, 40, 55, 97, 124, 235, 287, 482, 654, 1033, 1318, 2137, 2676, 4157, 5439, 7891, 10144, 15280, 19171, 27336, 35652, 49756, 63150, 89342, 111956, 154400, 197413, 264572, 336082, 456724, 568932, 756065, 959566, 1261803, 1576355, 2078267
Offset: 0

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(5) = 16 pairs of integer partitions:
      (51)|(6)
      (42)|(6)
     (411)|(6)
      (33)|(6)
     (321)|(6)
    (3111)|(6)
     (222)|(6)
     (222)|(33)
    (2211)|(6)
    (2211)|(33)
   (21111)|(6)
   (21111)|(33)
  (111111)|(6)
  (111111)|(42)
  (111111)|(33)
  (111111)|(222)

Crossrefs

Cf. A265947, A317144, A318915, A322435, A322436, A322439.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-i, min(n-i, i))*b(n, i+1), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018
  • Mathematica
    Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n-1} A026820(n, k) * A026794(n, k + 1).

A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

Original entry on oeis.org

1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

Examples

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)     (7)
            (21)   (31)  (32)     (42)    (43)
            (111)        (41)     (51)    (52)
                         (221)    (222)   (61)
                         (311)    (411)   (322)
                         (2111)   (2211)  (331)
                         (11111)          (421)
                                          (511)
                                          (2221)
                                          (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).

Crossrefs

The odd bisection is A058695.
The version for compositions is A213173.
The complement is counted by A322439 aerated.
The even bisection is A362051.
For mean instead of median we have A362559.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]

A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

Original entry on oeis.org

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Even bisection of A362558.
a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Examples

			The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Crossrefs

The version for compositions is A000302, bisection of A213173.
The complement is counted by A322439.
Even bisection of A362558.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with all equal run-sums.
A325347 counts partitions with integer median, complement A307683.
A353836 counts partitions by number of distinct run-sums.
A359893/A359901/A359902 count partitions by median.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]

A319884 Number of unordered pairs of set partitions of {1,...,n} where every block of one is a proper subset or proper superset of some block of the other.

Original entry on oeis.org

1, 0, 1, 7, 50, 481, 5667, 78058, 1238295, 22314627, 451354476, 10148011215, 251584513215, 6831141750512, 201976943666357, 6470392653260939, 223595676728884394, 8302299221314559877, 330075531021130110015, 14006780163088113914026, 632606447496264724088803
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2018

Keywords

Examples

			The a(3) = 7 pairs of set partitions:
  (1)(2)(3)|(123)
    (1)(23)|(12)(3)
    (1)(23)|(13)(2)
    (1)(23)|(123)
    (12)(3)|(13)(2)
    (12)(3)|(123)
    (13)(2)|(123)

Links

Crossrefs

Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A322435, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
costabstrQ[s_,t_]:=And@@Cases[s,x_:>Select[t,x!=#&&(SubsetQ[x,#]||SubsetQ[#,x])&]!={}];
Table[Length[Select[Subsets[sps[Range[n]],{2}],And[costabstrQ@@#,costabstrQ@@Reverse[#]]&]],{n,5}]
  • PARI
    F(x)={my(bell=(exp(y*(exp(x) - 1))  )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( 1 + exp( 2*(exp(exp(x) - 1) - exp(x)) ) * F(x) )/2)} \\ Andrew Howroyd, Jan 19 2024
  • PARI
    \\ 2nd prog, following formula - slightly slower
D(n,y) = (exp(2*y)/(1 + y)^2) * sum(k=0,n, x^k*sum(j=0, k, stirling(k,j,2) * y^j)^2/k!, O(x*x^n))
seq(n) = Vec(serlaplace((1/2)*(1 + D(n, exp(exp(x + O(x*x^n)) - 1) - 1)))) \\ Andrew Howroyd, Jan 20 2024

Formula

E.g.f.: (1/2)*(1 + D(x, exp(exp(x) - 1) - 1) ) where D(x,y) = (exp(2*y)/(1 + y)^2) * Sum_{k>=0} x^k*(Sum_{j=0..k} Stirling2(k,j)*y^j)^2/k!. - Andrew Howroyd, Jan 20 2024

Extensions

a(8) onwards from Andrew Howroyd, Jan 19 2024
Showing 1-10 of 11 results. Next

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