A087897 -id:A087897 - cp's OEIS frontend

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 4, 0, 1, 4, 1, 2, 6, 1, 1, 6, 3, 1, 8, 2, 1, 13, 1, 0, 13, 1, 7, 15, 1, 1, 19, 6, 1, 25, 1, 2, 33, 1, 1, 32, 5, 10, 39, 2, 1, 46, 14, 6, 55, 1, 1, 77, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Allowing even parts gives A018783, complement A000837.
For parts > 1 instead of gcd > 1 we have A087897.
For gcd = 1 instead of gcd > 1 we have A366843.
The strict case is A366750, with evens A303280.
The strict complement is A366844, with evens A078374.
A000041 counts integer partitions, strict A000009 (also into odd parts).
A000700 counts strict partitions into odd parts.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

More terms from Chai Wah Wu, Nov 02 2023

a(0)=0 prepended by Alois P. Heinz, Jan 11 2024

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A277643 Partial sums of number of overpartitions (A015128).

Original entry on oeis.org

1, 3, 7, 15, 29, 53, 93, 157, 257, 411, 643, 987, 1491, 2219, 3259, 4731, 6793, 9657, 13605, 19005, 26341, 36245, 49533, 67261, 90789, 121855, 162679, 216087, 285655, 375903, 492527, 642671, 835283, 1081539, 1395347, 1793987, 2298873, 2936465, 3739401, 4747849

Offset: 0

Vaclav Kotesovec, Oct 25 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A000070, A036469, A026906.

Cf. A211971, A002865, A087897.

Accumulate[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]]
nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)

a(n) = Sum_{k=0..n} A015128(k).
a(n) ~ exp(Pi*sqrt(n))/(4*Pi*sqrt(n)) * (1 + Pi/(4*sqrt(n))).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^k) / (1 - x^k). - Vaclav Kotesovec, Mar 25 2017
G.f.: 1/((1 - x)*theta_4(x)), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

A323087 Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.

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, 5, 1, 2, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(60) = 9 factorizations:
  (2*3*10), (2*5*6), (3*4*5),
  (2*30), (3*20), (4*15), (5*12), (6*10),
  (60).

Cf. A001597, A007916, A025147, A045778, A052410, A087897, A275972, A305150, A323054, A323086, A323090, A323091.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[strfacs[n],stableQ[#,IntegerQ[Log[#1,#2]]&]&]],{n,100}]

A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 7, 9, 11, 11, 15, 16, 18, 22, 24, 27, 32, 34, 41, 45, 51, 59, 64, 75, 82, 94, 105, 116, 132, 146, 163, 183, 202, 225, 251, 277, 309, 341, 378, 417, 463, 510, 564, 622, 685, 754, 830, 914, 1001, 1103, 1207, 1325

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			A list of all strict integer partitions using numbers that are not perfect powers begins:
   2: (2)        11: (6,3,2)    15: (13,2)       17: (12,5)
   3: (3)        12: (12)       15: (12,3)       17: (12,3,2)
   5: (5)        12: (10,2)     15: (10,5)       17: (11,6)
   5: (3,2)      12: (7,5)      15: (10,3,2)     17: (10,7)
   6: (6)        12: (7,3,2)    15: (7,6,2)      17: (10,5,2)
   7: (7)        13: (13)       15: (7,5,3)      17: (7,5,3,2)
   7: (5,2)      13: (11,2)     16: (14,2)       18: (18)
   8: (6,2)      13: (10,3)     16: (13,3)       18: (15,3)
   8: (5,3)      13: (7,6)      16: (11,5)       18: (13,5)
   9: (7,2)      13: (6,5,2)    16: (11,3,2)     18: (13,3,2)
   9: (6,3)      14: (14)       16: (10,6)       18: (12,6)
  10: (10)       14: (12,2)     16: (7,6,3)      18: (11,7)
  10: (7,3)      14: (11,3)     16: (6,5,3,2)    18: (11,5,2)
  10: (5,3,2)    14: (7,5,2)    17: (17)         18: (10,6,2)
  11: (11)       14: (6,5,3)    17: (15,2)       18: (10,5,3)
  11: (6,5)      15: (15)       17: (14,3)       18: (7,6,5)

Cf. A001597, A007916, A025147, A052410, A087897, A305631, A321346, A323054, A323089, A323090.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&FreeQ[#,1]&&And@@Not/@perpowQ/@#&]],{n,20}]

O.g.f.: Product_{n in A007916} (1 + x^n).

A226541 Number of unimodal compositions of n where the maximal part appears three times.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 3, 5, 7, 11, 16, 24, 34, 51, 71, 102, 143, 201, 276, 384, 522, 714, 964, 1301, 1739, 2328, 3084, 4085, 5377, 7064, 9226, 12036, 15616, 20228, 26092, 33584, 43067, 55125, 70308, 89502, 113598, 143889, 181755, 229160, 288186, 361750, 453046, 566346, 706464

Offset: 0

Joerg Arndt, Jun 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006330 (max part appears once), A114921 (max part appears twice).
Cf. A188674 (max part m appears m times), A001522 (max part m appears at least m times).
Cf. A001523 (max part appears any number of times).
Cf. A000009 (symmetric, max part m appears once; also symmetric, max part appears an odd number of times).
Cf. A035363 (symmetric, max part m appears twice; also symmetric, max part appears an even number of times).
Cf. A087897 (symmetric, max part m appears 3 times).
Cf. A027349 (symmetric, max part m appears m times), A189357 (symmetric, max part m appears at least m times).
Column k=3 of A247255.

N=66; x='x+O('x^N); Vec(sum(n=0,N, x^(3*n) / prod(k=1,n-1, 1-x^k )^2 ))

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n-1, 1-x^k )^2 ); replace 3 by m to obtain g.f. for "... max part appears m times".

a(n) ~ Pi^2 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Oct 24 2018

A341447 Heinz numbers of integer partitions whose only even part is the smallest.

Original entry on oeis.org

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317

Offset: 1

Gus Wiseman, Feb 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)

These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
The odd version is A341446.
A000009 counts partitions into odd parts, ranked by A066208.
A001222 counts prime factors.
A005843 lists even numbers.
A026804 counts partitions whose least part is odd, ranked by A340932.
A026805 counts partitions whose least part is even, ranked by A340933.
A027187 counts partitions with even length/max, ranked by A028260/A244990.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058696 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 4, 4, 5, 7, 10, 8, 13, 13, 15, 19, 25, 24, 35, 35, 43, 50, 61, 59, 79, 83, 98, 111, 137, 137, 176, 187, 219, 240, 284, 298, 360, 385, 444, 485, 568, 600, 706, 763, 867, 951, 1088, 1168, 1345, 1453, 1641, 1792, 2023, 2179, 2467, 2673, 2988

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(10) = 7 partitions:
  1  11  111  211   221    21111   2221     22211     22221      33211
              1111  2111   111111  22111    221111    32211      222211
                    11111          211111   2111111   2211111    322111
                                   1111111  11111111  21111111   2221111
                                                      111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(9) = 5 through a(12) = 8 partitions:
  (22221)      (33211)       (33221)        (2222211)
  (32211)      (222211)      (222221)       (3222111)
  (2211111)    (322111)      (322211)       (3321111)
  (21111111)   (2221111)     (332111)       (32211111)
  (111111111)  (22111111)    (2222111)      (222111111)
               (211111111)   (3221111)      (2211111111)
               (1111111111)  (22211111)     (21111111111)
                             (221111111)    (111111111111)
                             (2111111111)
                             (11111111111)

For parts instead of multiplicities we have A087897, complement A000009.
For multisets instead of partitions we have A105039, complement A363224.
The complement is counted by A363264.
For mode we have A363484, complement A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]

A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 3, 2, 7, 5, 9, 12, 13, 13, 22, 19, 29, 33, 39, 43, 63, 63, 82, 94, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

For parts instead of multiplicities we have A000009, complement A087897.
For multisets instead of partitions we have A363224, complement A105039.
The complement is counted by A363263.
For mode we have A363485, complement A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A067029 counts minima in prime factorization, co-modes A362613.
A071178 counts maxima in prime factorization, modes A362611.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]

A380412 First term of the n-th differences of the strict partition numbers. Inverse zero-based binomial transform of A000009.

Original entry on oeis.org

1, 0, 0, 1, -3, 7, -14, 25, -41, 64, -100, 165, -294, 550, -1023, 1795, -2823, 3658, -2882, -2873, 20435, -62185, 148863, -314008, 613957, -1155794, 2175823, -4244026, 8753538, -19006490, 42471787, -95234575, 210395407, -453413866, 949508390, -1931939460

Offset: 0

Gus Wiseman, Feb 03 2025

sign

Up to sign, same as A293467.

The version for non-strict partitions is A281425, row n=0 of A175804.
Column n=0 of A378622.
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of A000009, differences A129519.
Cf. A000041, A007442, A008284, A218482, A293467, A320590, A320591, A377285, A378970.

nn=10;Table[First[Differences[PartitionsQ/@Range[0,nn],n]],{n,0,nn}]

a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000041(k).

cp's OEIS Frontend

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

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A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A277643 Partial sums of number of overpartitions (A015128).

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A323087 Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.

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A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

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A226541 Number of unimodal compositions of n where the maximal part appears three times.

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A341447 Heinz numbers of integer partitions whose only even part is the smallest.

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A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

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A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

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