A018783 -id:A018783 - cp's OEIS frontend

A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

0, 0, 1, 1, 4, 5, 9, 13, 21, 29, 43, 56, 79, 109, 146, 192, 254, 329, 428, 553, 707, 900, 1139, 1434, 1800, 2251, 2799, 3472, 4286, 5275, 6469, 7918, 9655, 11755, 14252, 17248, 20817, 25084, 30134, 36142, 43235, 51644, 61548, 73241, 86961, 103108, 122010

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
The a(8) = 13 partitions:
  (62),
  (332), (422), (431), (521), (611),
  (3221), (4211),
  (22211), (32111), (41111),
  (221111),
  (2111111).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000837, A007359, A018783, A051424, A281116, A289508, A289509, A318978.

Table[Length[Select[IntegerPartitions[n],GCD@@(#+1)==1&]],{n,30}]

seq(n)={Vec(sum(d=1, n+1, moebius(d)*(-1 + 1/prod(k=ceil(2/d), (n+1)\d, 1 - x^(k*d-1) + O(x*x^n)))), -n)} \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - Andrew Howroyd, Oct 17 2019

A366850 Number of integer partitions of n whose odd parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 32, 43, 60, 80, 110, 140, 194, 244, 327, 410, 544, 670, 883, 1081, 1401, 1708, 2195, 2651, 3382, 4069, 5129, 6157, 7708, 9194, 11438, 13599, 16788, 19911, 24432, 28858, 35229, 41507, 50359, 59201, 71489, 83776, 100731, 117784

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (53)
             (111)  (211)   (221)    (321)     (331)      (71)
                    (1111)  (311)    (411)     (421)      (431)
                            (2111)   (2211)    (511)      (521)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For all parts (not just odd) we have A000837, complement A018783.
The complement is counted by A366842.
These partitions have ranks A366846.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.

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

A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 324, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 2916, 3600, 4096, 5184, 5832, 7776, 8100, 8192, 9216, 11664, 13824, 14400, 16384, 20736, 22500, 26244, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 72900

Offset: 1

Gus Wiseman, May 13 2018

nonn

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and periodic, meaning its multiplicities have a common divisor greater than 1.

			Sequence of all normal periodic multisets begins
4:    {1,1}
8:    {1,1,1}
16:   {1,1,1,1}
32:   {1,1,1,1,1}
36:   {1,1,2,2}
64:   {1,1,1,1,1,1}
128:  {1,1,1,1,1,1,1}
144:  {1,1,1,1,2,2}
216:  {1,1,1,2,2,2}
256:  {1,1,1,1,1,1,1,1}
324:  {1,1,2,2,2,2}
512:  {1,1,1,1,1,1,1,1,1}
576:  {1,1,1,1,1,1,2,2}
900:  {1,1,2,2,3,3}
1024: {1,1,1,1,1,1,1,1,1,1}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000837, A000961 A001597, A007916, A018783, A052409, A052410, A055932, A056239, A112798, A178472, A303431, A303709, A304450, A368682.

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]>1&]

Intersection of A001597 and A055932.

A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 2, 5, 5, 9, 5, 15, 9, 19, 16, 28, 16, 44, 21, 55, 38, 73, 34, 109, 46, 130, 73, 170, 66, 251, 78, 287, 137, 364, 119, 522, 135, 590, 236, 759, 190, 1042, 219, 1175, 425, 1460, 306, 2006, 347, 2277, 671, 2780, 471, 3734, 584, 4197, 1087

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(2) = 1 through a(12) = 15 partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)      (B)     (C)
            (22)       (33)   (52)  (44)    (63)   (55)     (83)    (66)
                       (42)         (62)    (72)   (64)     (92)    (84)
                       (222)        (422)   (333)  (73)     (722)   (93)
                                    (2222)  (522)  (82)     (5222)  (A2)
                                                   (442)            (444)
                                                   (622)            (552)
                                                   (4222)           (633)
                                                   (22222)          (642)
                                                                    (822)
                                                                    (3333)
                                                                    (4422)
                                                                    (6222)
                                                                    (42222)
                                                                    (222222)

The complement to these partitions is counted by A328164.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]!=GCD@@(#-1)&]],{n,0,30}]

A366846 Numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122

Offset: 1

Gus Wiseman, Oct 29 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The odd prime indices of 115 are {3,9}, and these are not relatively prime, so 115 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, and these are relatively prime, so 825 is in the sequence.

Including even indices gives A289509, ones of A289508, counted by A000837.
The complement when including even indices is A318978, counted by A018783.
The nonzero complement ranks the partitions counted by A366842.
The version for halved even indices is A366847.
The odd case is A366848.
The partitions with these Heinz numbers are counted by A366850.
A000041 counts integer partitions, strict A000009 (also into odds).
A112798 lists prime indices, length A001222, sum A056239.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A087436, A302696, A302697, A325698, A366843, A366844, A366849.

Select[Range[100], GCD@@Select[PrimePi/@First/@FactorInteger[#], OddQ]==1&]

A319810 Number of fully periodic integer partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 8, 7, 11, 2, 17, 2, 18, 9, 15, 2, 32, 5, 22, 12, 34, 2, 54, 2, 49, 16, 51, 10, 94, 2, 77, 23, 112, 2, 152, 2, 148, 47, 165, 2, 258, 7, 247, 52, 286, 2, 400, 17, 402, 78, 439, 2, 657, 2, 594, 131, 711, 24

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.

			The a(12) = 11 fully periodic integer partitions:
  (12)
  (6,6)
  (4,4,4)
  (5,5,1,1)
  (4,4,2,2)
  (3,3,3,3)
  (3,3,3,1,1,1)
  (3,3,2,2,1,1)
  (2,2,2,2,2,2)
  (2,2,2,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)
Periodic partitions missing from this list are:
  (4,4,1,1,1,1)
  (3,3,1,1,1,1,1,1)
  (2,2,2,1,1,1,1,1,1)
  (2,2,1,1,1,1,1,1,1,1)
The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).
The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.

totperQ[m_]:=Or[Length[m]==1,And[GCD@@Length/@Split[Sort[m]]>1,totperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],totperQ]],{n,30}]

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

A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 0, 37, 0, 44, 16, 56, 0, 93, 6, 101, 29, 137, 0, 217, 0, 230, 57, 297, 20, 450, 0, 490, 102, 643, 0, 918, 0, 1004, 202, 1255, 0, 1783, 14, 1992, 298, 2438, 0, 3364, 61, 3734, 491, 4565, 0, 6251, 0, 6842, 818

Offset: 1

Len Smiley, Jul 25 2005

nonn

The singleton and the all-ones partitions are ignored, so that a(n)=0 if n is prime. If a partition is listed as m_1^am_2^bm_3^c..., then it is counted exactly when gcd(a,b,c,...)>1. These are equinumerous (conjugate) with those partitions for which gcd(m_1,m_2,...)>1 (less 1, the singleton), hence the formula.

			a(25) = 6: 1^(15)2^5 = 5{1, 1, 1, 2}, 1^52^(10) = 5{1, 2, 2}, 1^(10)3^5 = 5{3, 1, 1}, 2^53^5 = 5{3, 2}, 1^44^4 = 5{4, 1}, 5^5 = 5{5}.
Note that A000041(25)=P(25)=1958, only 6 of which satisfy the criterion.

Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.

with(combinat):PartMulti:=proc(n::nonnegint) local count,a,i,j,b,m,k,part_vec;
bigcount:=0; if isprime(n) then return(bigcount) else ps:=partition(n); b:=nops(ps);
for m from 2 to b-1 do p:=ps[m]; a:=nops(p); part_vec:=array(1..n);
for k from 1 to n do part_vec[k]:=0 od;
for i from 1 to a do j:=p[i]; part_vec[j]:=part_vec[j]+1 od;
g:=0; for j from 1 to n do g:=igcd(g,part_vec[j]) od;
if g>1 then bigcount:=bigcount+1 fi od; return(bigcount) end if end proc;
seq(PartMulti(q),q=1..49);

Table[Length[Select[IntegerPartitions[n],And[Length[#]1]&]],{n,20}] (* Gus Wiseman, Dec 06 2018 *)

a(n) = A018783(n)-1, n>1. - Vladeta Jovovic, Jul 28 2005

More terms from Gus Wiseman, Dec 06 2018

A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 13, 17, 25, 33, 51, 62, 92, 116, 160, 203, 281, 341, 469, 572, 754, 929, 1221, 1466, 1912, 2306, 2937, 3548, 4499, 5353, 6764, 8062, 10006, 11946, 14764, 17455, 21502, 25425, 30949, 36579, 44393, 52132, 63042, 74000, 88709, 104098, 124448

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(1) = 1 through a(8) = 17 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (61)       (71)
                    (1111)  (221)    (411)     (322)      (332)
                            (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (511)      (611)
                                     (111111)  (2221)     (3221)
                                               (3211)     (3311)
                                               (4111)     (4211)
                                               (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The complement to these partitions is counted by A328163.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]==GCD@@(#-1)&]],{n,0,30}]

A331885 Number of partitions of n into parts having a common factor > 1 with n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 16, 9, 22, 1, 51, 1, 51, 17, 57, 1, 147, 7, 102, 30, 152, 1, 620, 1, 231, 58, 298, 21, 946, 1, 491, 103, 921, 1, 3249, 1, 1060, 325, 1256, 1, 4866, 15, 3157, 299, 2539, 1, 10369, 62, 4846, 492, 4566, 1, 45786, 1, 6843

Offset: 0

Ilya Gutkovskiy, Jan 30 2020

nonn

			a(6) = 4 because we have [6], [4, 2], [3, 3] and [2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

Cf. A182986 (positions of 1's), A018783, A057562, A121998, A331887, A331888.

a:= proc(m) option remember; local b; b:=
      proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       `if`(igcd(i, m)>1, b(n-i, min(i, 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/(1 - Boole[GCD[k, n] > 1] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 62}]

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

cp's OEIS Frontend

A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

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A366850 Number of integer partitions of n whose odd parts are relatively prime.

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A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

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A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

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A366846 Numbers whose odd prime indices are relatively prime.

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A319810 Number of fully periodic integer partitions of n.

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

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A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

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