A303280 -id:A303280 - cp's OEIS frontend

A318717 Number of strict integer partitions of n in which no two parts are relatively prime.

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 2, 32, 13, 38, 7, 57, 2, 54, 19, 68, 3, 95, 3, 90, 33, 104, 3, 148, 7, 149, 40, 166, 5, 230, 17, 226, 56, 256, 6, 360, 9, 340, 84, 390, 25, 527, 11, 513, 109

Offset: 0

Gus Wiseman, Sep 02 2018

nonn

			The a(20) = 11 partitions:
  (20),
  (12,8), (14,6), (15,5), (16,4), (18,2),
  (10,6,4), (10,8,2), (12,6,2), (14,4,2),
  (8,6,4,2).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

Cf. A000009, A007359, A007360, A051185, A302569, A302797, A303140, A303280, A303283, A305713, A305843, A305854, A305713, A318715, A318719.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]]&]],{n,30}]

a(51)-a(69) from Alois P. Heinz, Sep 02 2018

A318716 Heinz numbers of strict integer partitions with relatively prime parts in which no two parts are relatively prime.

Original entry on oeis.org

2, 17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847, 303949

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of strict integer partitions with Heinz numbers in the sequence begins: (1), (15,10,6), (21,14,6), (20,15,6), (15,12,10), (45,10,6), (18,15,10).

Cf. A078374, A289509, A302569, A302696, A302796, A302797, A303140, A303280, A303282, A303283, A305713, A318715, A318718, A318719.

Select[Range[100000],With[{m=PrimePi/@FactorInteger[#][[All,1]]},And[SquareFreeQ[#],GCD@@m==1,And@@(GCD[##]>1&)@@@Select[Tuples[m,2],Less@@#&]]]&]

A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1

Offset: 1

Gus Wiseman, Apr 19 2018

			Triangle begins:
01:   1
02:   0  1
03:   1  0  1
04:   1  0  0  1
05:   2  0  0  0  1
06:   2  1  0  0  0  1
07:   4  0  0  0  0  0  1
08:   4  1  0  0  0  0  0  1
09:   6  0  1  0  0  0  0  0  1
10:   7  2  0  0  0  0  0  0  0  1
11:  11  0  0  0  0  0  0  0  0  0  1
12:  10  2  1  1  0  0  0  0  0  0  0  1
13:  17  0  0  0  0  0  0  0  0  0  0  0  1
14:  17  4  0  0  0  0  0  0  0  0  0  0  0  1
15:  23  0  2  0  1  0  0  0  0  0  0  0  0  0  1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2:  (10,2) (6,4,2)
T(12,3) = 1:  (9,3)
T(12,4) = 1:  (8,4)
T(12,12) = 1: (12)

First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.

Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]

If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.

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

Original entry on oeis.org

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

A331887 Number of partitions of n into distinct parts having a common factor > 1 with n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 11, 1, 11, 6, 12, 1, 23, 3, 18, 8, 23, 1, 69, 1, 32, 13, 38, 7, 84, 1, 54, 19, 79, 1, 224, 1, 90, 46, 104, 1, 264, 5, 187, 39, 166, 1, 449, 14, 251, 55, 256, 1, 1374, 1, 340, 111, 390, 20, 1692, 1, 513, 105, 1610

Offset: 0

Ilya Gutkovskiy, Jan 30 2020

nonn

			a(12) = 5 because we have [12], [10, 2], [9, 3], [8, 4] and [6, 4, 2].

Antti Karttunen, Table of n, a(n) for n = 0..10416
Index entries for sequences related to partitions

Cf. A036998, A121998, A175787 (positions of 1's), A303280, A331885, A331888.

a:= proc(m) option remember; local b; b:=
      proc(n, i) option remember; `if`(i*(i+1)/21, b(n-i, min(i-1, n-i)), 0)+b(n, i-1)))
      end; forget(b); b(m$2)
    end:
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 30 2020

Table[SeriesCoefficient[Product[(1 + Boole[GCD[k, n] > 1] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 70}]

A331887(n) = { my(p = Ser(1, 'x, 1+n)); for(k=2, n, if(gcd(n,k)>1, p *= (1 + 'x^k))); polcoef(p, n); }; \\ Antti Karttunen, Jan 25 2025

a(n) = [x^n] Product_{k: gcd(n,k) > 1} (1 + x^k).

A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A000009, A000837, A051424, A078374, A289509, A302696, A302796, A302797, A303280, A305713.

Select[Range[200],And[SquareFreeQ[#],GCD@@PrimePi/@FactorInteger[#][[All,1]]>1]&]

A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 4, 1, 4, 1, 0, 1, 6, 1, 1, 7, 2, 0, 1, 11, 1, 10, 2, 1, 1, 0, 1, 17, 1, 17, 4, 0, 1, 23, 2, 1, 1, 26, 4, 1, 0, 1, 37, 1, 36, 6, 2, 1, 0, 1, 53, 1, 53, 7, 2, 1, 0, 1, 70, 4, 1, 1, 77, 11, 0, 1, 103, 1, 103, 10, 4, 2, 1

Offset: 1

Gus Wiseman, Sep 16 2018

			Triangle begins:
   1
   0  1
   1  1
   1  0  1
   2  1
   2  1  0  1
   4  1
   4  1  0  1
   6  1  1
   7  2  0  1
  11  1
  10  2  1  1  0  1
  17  1
  17  4  0  1
  23  2  1  1
  26  4  1  0  1
  37  1
  36  6  2  1  0  1
  53  1
  53  7  2  1  0  1

A regular version is A303138. Row lengths are A000005. Row sums are A000009. First column is A078374.

Cf. A000837, A027750, A289508, A289509, A303140, A303280, A319299, A319301.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==k]&]],{n,20},{k,Divisors[n]}]

T(n,k) = A078374(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

Gus Wiseman, Nov 07 2020

nonn

			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)

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.

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

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

Gus Wiseman, Nov 01 2023

nonn

			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)

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.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]

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

More terms from Chai Wah Wu, Nov 02 2023

cp's OEIS Frontend

A318717 Number of strict integer partitions of n in which no two parts are relatively prime.

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A318716 Heinz numbers of strict integer partitions with relatively prime parts in which no two parts are relatively prime.

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A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

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

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A331887 Number of partitions of n into distinct parts having a common factor > 1 with n.

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A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

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A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

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

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

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