A168532 -id:A168532 - cp's OEIS frontend

A168534 Triangle read by rows, A168532 * A000012; as infinite lower triangular matrices.

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 5, 2, 2, 1, 1, 1, 1, 30, 3, 3, 1, 1, 1, 1, 1, 1, 42, 8, 2, 2, 2, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 28 2009

Row sums = A078392: (1, 3, 5, 9, 11, 20, 21,...).
Triangle A168533 = A000012 * A168532
Left border = the partition numbers, A000041 starting with offset 1.

			First few rows of the triangle =
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 1, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 5, 2, 2, 1, 1, 1, 1;
30, 3, 3, 1, 1, 1, 1, 1, 1;
42, 8, 2, 2, 2, 1, 1, 1, 1, 1;
56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
77, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1;
101, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
135, 16, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
176, 9, 9, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
231, 22, 5, 5, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A168532, A168533, A000041, A078392

Triangle read by rows, A168532 * A000012; where A000012 = an infinite lower
triangular matrix with all 1's. The operation takes partial row sums
starting from the right of each row.

A168533 Triangle read by rows, A000012 * A168532, as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 2, 1, 1, 13, 2, 1, 1, 1, 20, 4, 2, 1, 1, 1, 34, 4, 2, 1, 1, 1, 1, 51, 7, 2, 2, 1, 1, 1, 1, 78, 7, 4, 2, 1, 1, 1, 1, 1, 112, 13, 4, 2, 2, 2, 21, 1, 1, 167, 13, 4, 2, 2, 1, 1, 1, 1, 1, 1, 230, 20, 7, 4, 2, 2, 21, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 28 2009

Row sums = A026905: (1, 3, 6, 11, 18, 29, 44, 66,...)

Triangle A168534 = A168532 * A000012

			First few rows of the triangle =
1;
2, 1;
4, 1, 1;
7, 2, 1, 1;
13, 2, 1, 1, 1;
20, 4, 2, 1, 1, 1;
34, 4, 2, 1, 1, 1, 1;
51, 7, 2, 2, 1, 1, 1, 1;
78, 7, 4, 2, 1, 1, 1, 1, 1;
112, 13, 4, 2, 2, 1, 1, 1, 1, 1;
167, 13, 4, 2, 2, 1, 1, 1, 1, 1, 1;
230, 20, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1;
330, 20, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1;
449, 34, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
616, 34, 13, 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
825, 51, 13, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
1121, 51, 13, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Cf. A168532, A026905, A168534

Triangle read by rows, A000012 * A168532; where A000012 = an infinite lower
triangular matrix with all 1's. The operation takes partial sums of A168532
column terms.

A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 100, 119, 167, 209, 296, 347, 489, 582, 775, 945, 1254, 1481, 1951, 2334, 2980, 3580, 4564, 5386, 6841, 8118, 10085, 12012, 14862, 17526, 21636, 25524, 31082, 36694, 44582, 52255, 63260, 74170, 88931, 104302

Offset: 0

N. J. A. Sloane

Starting (1, 1, 2, 3, 6, 7, 14, ...), = row sums of triangle A137585. - Gary W. Adamson, Jan 27 2008
Triangle A168532 has aerated variants of this sequence in each column starting with offset 1, row sums = A000041. -  Gary W. Adamson, Nov 28 2009
A partition is aperiodic iff its multiplicities are relatively prime, i.e., its Heinz number (A215366) is not a perfect power (A007916). - Gus Wiseman, Dec 19 2017
This sequence is monotonically increasing; each partition of n-1 can have a part of size 1 added to it to get a partition counted in a(n). - Franklin T. Adams-Watters, Jul 24 2020

			Of the 11 partitions of 6, we must exclude 6, 4+2, 3+3 and 2+2+2, so a(6) = 11 - 4 = 7.
For n=6, 2+2+1+1 is periodic because it can be written 2*(2+1), similarly 1+1+1+1+1+1, 3+3 and 2+2+2.
The a(6) = 7 partitions into relatively prime parts are (51), (411), (321), (3111), (2211), (21111), (111111). The a(6) = 7 aperiodic partitions are (6), (51), (42), (411), (321), (3111), (21111). - _Gus Wiseman_, Dec 19 2017

H. W. Gould, personal communication.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Mohamed El Bachraoui, On the Parity of p(n,3) and p_psi(n,3), Contributions to Discrete Mathematics, Vol. 5.2 (2010).
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Remark 3.4.
N. J. A. Sloane, Transforms

Cf. A000041, A018783.

Cf. A000740, A007916, A047968, A055892, A100953, A137585, A168532, A281116.

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; g[n_, j_] := Apply[GCD, Part[p[n], j]]; h[n_] := Table[g[n, j], {j, 1, l[n]}]; Join[{1}, Table[Count[h[n], 1], {n, 1, 20}]]
(* Clark Kimberling, Mar 09 2012 *)
a[0] = 1; a[n_] := Sum[ MoebiusMu[n/d] * PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 03 2013 *)

N=66; x='x+O('x^N); gf=2+sum(n=1,N, (1/eta(x^n))*moebius(n)); Vec(gf) \\ Joerg Arndt, May 11 2013

print1("1, "); for(n=1,46,my(s=0);forpart(X=n,s+=gcd(X)==1);print1(s,", ")) \\ Hugo Pfoertner, Mar 27 2020

from sympy import npartitions, mobius, divisors
def a(n): return 1 if n==0 else sum(mobius(n//d)*npartitions(d) for d in divisors(n)) # Indranil Ghosh, Apr 26 2017

Möbius transform of A000041. - Christian G. Bower, Jun 11 2000
Product_{n>0} 1/(1-q^n) = 1 + Sum_{n>0} a(n)*q^n/(1-q^n). - Mamuka Jibladze, Nov 14 2015
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019
a(n) <= p(n) <= a(n+1), where p(n) is the number of partitions of n (A000041). - Franklin T. Adams-Watters, Jul 24 2020

Corrected and extended by David W. Wilson, Aug 15 1996

Additional name from Christian G. Bower, Jun 11 2000

A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.

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

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

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

from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A366843 Number of integer partitions of n into odd, relatively prime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 21, 23, 32, 37, 42, 53, 62, 70, 88, 103, 116, 139, 164, 184, 220, 255, 283, 339, 390, 435, 511, 578, 653, 759, 863, 963, 1107, 1259, 1401, 1609, 1814, 2015, 2303, 2589, 2878, 3259, 3648, 4058, 4580, 5119, 5672, 6364

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (3111)    (511)      (71)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Allowing even parts gives A000837.
The strict case is A366844, with evens A078374.
The complement is counted by A366852, with evens A018783.
The pairwise coprime version is A366853, with evens A051424.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A113685, A116598, A289509, A289508, A302697, A337485, A366845, A366848, A366849.

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

from math import gcd
from sympy.utilities.iterables import partitions
def A366843(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 0, 0, 1, 0, 1, 1, 4, 2, 4, 1, 5, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 0, 4, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 6, 13, 17, 33, 37, 68, 82, 125, 159, 237, 278, 409, 491, 674, 830, 1121, 1329, 1781, 2144, 2770, 3345, 4299, 5086, 6507, 7752, 9687, 11571, 14378, 16985, 21039, 24876, 30379, 35924, 43734, 51320, 62238, 73068, 87747, 103021, 123347, 143955

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(7) = 5 partitions are (421), (331), (322), (2221), (22111).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303140.

Table[Select[IntegerPartitions[n],!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,30}]

A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

Original entry on oeis.org

0, 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, 1, 32, 13, 38, 7, 57, 1, 54, 19, 68, 1, 95, 1, 90, 33, 104, 1, 148, 5, 149, 39, 166, 1, 230, 14, 226, 55, 256, 1, 360, 1, 340, 82, 390, 20, 527, 1, 513, 105, 609, 1

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

			The a(18) = 10 strict partitions are (18), (10,8), (12,6), (14,4), (15,3), (16,2), (8,6,4), (9,6,3), (10,6,2), (12,4,2).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000009, A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303138.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 23 2018

Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d],{d,Rest[Divisors[n]]}],{n,100}]

a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).

cp's OEIS Frontend

A168534 Triangle read by rows, A168532 * A000012; as infinite lower triangular matrices.

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A168533 Triangle read by rows, A000012 * A168532, as infinite lower triangular matrices.

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A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions.

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A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

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A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

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A366843 Number of integer partitions of n into odd, relatively prime parts.

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A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

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A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.