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.

Previous Showing 61-67 of 67 results.

A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254
Offset: 1

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Examples

			Triangle begins:
    1
    1   1
    2   1
    3   1   1
    6   1
    7   2   1   1
   14   1
   17   3   1   1
   27   2   1
   34   6   1   1
   55   1
   63   7   3   2   1   1
  100   1
  119  14   1   1
  167   6   2   1
  209  17   3   1   1
  296   1
  347  27   7   2   1   1
  489   1
  582  34   6   3   1   1

Links

Crossrefs

A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.
Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.

Programs

  • Maple
    # with table A000837 obtained from that sequence
f:= proc(n) local D,d;
  D:= sort(convert(numtheory:-divisors(n),list),`>`);
  seq(A000837[d],d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],GCD@@#==k&]],{n,20},{k,Divisors[n]}]

Formula

T(n,k) = A000837(n/A027750(n,k)).

A338554 Number of non-constant integer partitions of n whose parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 0, 9, 0, 13, 6, 18, 0, 33, 0, 40, 14, 54, 0, 87, 5, 99, 27, 133, 0, 211, 0, 226, 55, 295, 18, 443, 0, 488, 100, 637, 0, 912, 0, 1000, 198, 1253, 0, 1775, 13, 1988, 296, 2434, 0, 3358, 59, 3728, 489, 4563, 0, 6241, 0, 6840, 814
Offset: 0

Views

Author

Gus Wiseman, Nov 07 2020

Keywords

Examples

			The a(6) = 2 through a(15) = 6 partitions (empty columns indicated by dots, A = 10, B = 11, C = 12):
  (42)  .  (62)   (63)  (64)    .  (84)     .  (86)      (96)
           (422)        (82)       (93)        (A4)      (A5)
                        (442)      (A2)        (C2)      (C3)
                        (622)      (633)       (644)     (663)
                        (4222)     (642)       (662)     (933)
                                   (822)       (842)     (6333)
                                   (4422)      (A22)
                                   (6222)      (4442)
                                   (42222)     (6422)
                                               (8222)
                                               (44222)
                                               (62222)
                                               (422222)

Crossrefs

A046022 lists positions of zeros.
A082023(n) - A059841(n) is the 2-part version, n > 2.
A303280(n) - 1 is the strict case (n > 1).
A338552 lists the Heinz numbers of these partitions.
A338553 counts the complement, with Heinz numbers A338555.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A018783 counts non-relatively prime partitions (ordered: A178472), with Heinz numbers A318978.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A008284, A051424, A082024, A289508, A302698, A304712.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&GCD@@#>1&]],{n,0,30}]

Formula

For n > 0, a(n) = A018783(n) - A000005(n) + 1.

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 2, 0, 1, 3, 1, 0, 3, 2, 1, 4, 1, 1, 5, 0, 1, 5, 1, 2, 5, 1, 1, 5, 2, 2, 6, 0, 1, 9, 1, 0, 9, 0, 3, 9, 1, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)

Crossrefs

This is the case of A000700 with a common divisor.
Including evens gives A303280.
The complement is counted by A366844, non-strict version A366843.
The non-strict version is A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366750(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

Extensions

More terms from Chai Wah Wu, Nov 02 2023

A202523 Number of partitions of n into distinct parts having pairwise prime GCDs but no overall common factor.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 1, 0, 4, 0, 1, 0, 5, 0, 4, 0, 0, 0, 3, 0, 5, 0, 0, 0, 6, 0, 1, 1, 2, 0, 6, 0, 6, 1, 1, 0, 4, 0, 12, 0, 1, 1, 12, 1, 9, 0, 0, 1, 10, 0, 10, 0, 1, 2, 10, 1, 4
Offset: 31

Views

Author

Alois P. Heinz, Dec 20 2011

Keywords

Examples

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(37) = 1: [10,12,15] = [2*5,2*2*3,3*5].
a(41) = 2: [6,15,20], [6,14,21].
a(43) = 1: [10,15,18].
a(47) = 1: [12,14,21].
a(49) = 1: [10,15,24].
a(61) = 4: [6,22,33], [10,15,36], [6,15,40], [6,10,45].

Links

Crossrefs

Cf. A202385, A202425, A200976, A018783.

Programs

  • Maple
    w:=(m, h)-> mul(`if`(j[1]>=h, 1, j[1]^min(j[2], 2)), j=ifactors(m)[2]):
b:= proc(n, i, g, s) option remember; local j, ok, si;
      if n=0 then `if`(g>1, 0, 1)
    elif i<2 or member(1, s) then 0
    else ok:= evalb(i<=n);
         si:= map(x->w(x, i), s);
         for j in s while ok do ok:= isprime(igcd(i, j)) od;
         b(n, i-1, g, si) +`if`(ok,
         b(n-i, i-1, igcd(i, g), si union {w(i, i)} ), 0)
      fi
    end:
a:= n-> b(n, n, 0, {}):
seq(a(n), n=31..100);
  • Mathematica
    w[m_, h_] := Product[If[j[[1]] >= h, 1, j[[1]]^Min[j[[2]], 2]], {j, FactorInteger[m]}];
b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n == 0, If[g > 1, 0, 1], i < 2 || MemberQ[s, 1], 0, True, ok = i <= n; si = w[#, i]& /@ s; For[j = 1, j <= Length[s], j++, If[!ok, Break[]]; ok = PrimeQ[ GCD[i, s[[j]]]]]; b[n, i - 1, g, si] + If[ok, b[n - i, i - 1, GCD[i, g], si ~Union~ {w[i, i]}], 0]]];
a[n_] := b[n, n, 0, {}];
Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

A304648 Number of different periodic multisets that fit within some normal multiset of weight n.

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 44, 78, 136, 242, 422, 747, 1314, 2326, 4121, 7338, 13052, 23288, 41568, 74329, 133011, 238338, 427278, 766652, 1376258, 2472012, 4441916, 7984990, 14358424, 25826779, 46465956, 83616962, 150497816, 270917035, 487753034, 878244512
Offset: 1

Views

Author

Gus Wiseman, May 15 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. It is periodic if its multiplicities have a common divisor greater than 1.

Examples

			The a(5) = 13 periodic multisets:
(11), (22), (33), (44),
(111), (222), (333),
(1111), (1122), (1133), (2222), (2233),
(11111).

Links

Crossrefs

Cf. A000217, A001597, A018783, A027941, A178472, A210554, A303547, A303709, A303974, A303976.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]
  • PARI
    seq(n)=Vec(sum(d=2, n, -moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x), -n) \\ Andrew Howroyd, Feb 04 2021

Formula

From Andrew Howroyd, Feb 04 2021: (Start)
a(n) = A027941(n) - A303976(n).
G.f.: Sum_{d>=2} -mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)).
(End)

Extensions

a(12)-a(13) from Robert Price, Sep 15 2018
Terms a(14) and beyond from Andrew Howroyd, Feb 04 2021

A305736 Number of integer partitions of n whose greatest common divisor is composite (nonprime and > 1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 4, 0, 1, 1, 5, 0, 4, 0, 8, 1, 1, 0, 14, 1, 1, 3, 16, 0, 10, 0, 22, 1, 1, 1, 41, 0, 1, 1, 45, 0, 18, 0, 57, 9, 1, 0, 94, 1, 8, 1, 102, 0, 38, 1, 138, 1, 1, 0, 221, 0, 1, 17, 231, 1, 59, 0, 298, 1, 22
Offset: 1

Views

Author

Gus Wiseman, Jun 22 2018

Keywords

Examples

			The a(12) = 4 integer partitions are (12), (8 4), (6 6), (4 4 4).

Links

Crossrefs

Cf. A000040, A000837, A018783, A100953, A289508, A302698, A305563, A305731, A305735.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!(GCD@@#==1||PrimeQ[GCD@@#])&]],{n,0,20}]
  • PARI
    seq(n)={dirmul(vector(n, n, numbpart(n)), dirmul(vector(n, n, moebius(n)), vector(n, n, n>1&&!isprime(n))))} \\ Andrew Howroyd, Jun 22 2018

Formula

a(n) = A018783(n) - A305735(n). - Andrew Howroyd, Jun 22 2018

A319811 Number of totally aperiodic integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 99, 117, 162, 203, 286, 333, 469, 558, 737, 903, 1196, 1414, 1860, 2232, 2839, 3422, 4359, 5144, 6531, 7762, 9617, 11479, 14182, 16715, 20630, 24333, 29569, 34890, 42335, 49515, 59871, 70042, 83810, 98105, 117152
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2018

Keywords

Comments

An integer partition is totally aperiodic iff either it is strict or it is aperiodic with totally aperiodic multiplicities.

Examples

			The a(6) = 7 aperiodic integer partitions are: (6), (51), (42), (411), (321), (3111), (21111). The first aperiodic integer partition that is not totally aperiodic is (432211).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319810.

Programs

  • Mathematica
    totaperQ[m_]:=Or[UnsameQ@@m,And[GCD@@Length/@Split[Sort[m]]==1,totaperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],totaperQ]],{n,30}]
Previous Showing 61-67 of 67 results.

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