A055922 -id:A055922 - cp's OEIS frontend

A358334 Number of twice-partitions of n into odd-length partitions.

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A366844 Number of strict integer partitions of n into odd relatively prime parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334

Offset: 0

Gus Wiseman, Oct 29 2023

nonn

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)

This is the relatively prime case of A000700.
The pairwise coprime version is the odd-part case of A007360.
Allowing even parts gives A078374.
The halved even version is A078374 aerated.
The non-strict version is A366843, with evens A000837.
The complement is counted by the strict case of 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.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, 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 A366844(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, Oct 30 2023

More terms from Chai Wah Wu, Oct 30 2023

A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

Original entry on oeis.org

1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230

Offset: 1

Gus Wiseman, Mar 18 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.

			The terms together with their prime indices begin:
   1 = 1
   2 = prime(1)
   5 = prime(3)
   8 = prime(1)^3
  10 = prime(1) prime(3)
  11 = prime(5)
  17 = prime(7)
  22 = prime(1) prime(5)
  23 = prime(9)
  31 = prime(11)
  32 = prime(1)^5
  34 = prime(1) prime(7)
  40 = prime(1)^3 prime(3)

David A. Corneth, Table of n, a(n) for n = 1..10000

The restriction to primes is A031368.
The first condition alone is A066208, counted by A000009.
These partitions are counted by A117958.
The squarefree case is A258116, even A258117.
The second condition alone is A268335, counted by A055922.
The even-even version is A352141 counted by A035444.
A000290 = exponents all even, counted by A035363.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.

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

from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())),count(max(startvalue,1)))
A352142_list = list(islice(A352142_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A066208 and A268335.
A257991(a(n)) = A001222(a(n)).
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A162641(a(n)) = 0.

A130219 Number of partitions of 2n-set in which number of blocks of size k is even (or zero) for every k.

Original entry on oeis.org

1, 1, 4, 56, 631, 15457, 582374, 18589286, 894499204, 51154344582, 3823359163826, 274722100927166, 25458967562911128, 2569179797929092506, 284554990016509385086, 37830153187190688287522, 5093072752898942262610007, 798814778335473578083666573

Offset: 0

Vladeta Jovovic, Aug 04 2007, Aug 05 2007

			a(2)=4 because we have ab|cd, ac|bd, ad|bc and a|b|c|d.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A055922, A102759, A111723, A111724.

g:=product(cosh(x^k/factorial(k)),k=1..35): gser:=series(g,x=0,32): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..14); # Emeric Deutsch, Sep 01 2007
# second Maple program:
g:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
     `if`(irem(j, 2)=0, g(n-i*j, i-1, p+j*i)/j!/i!^j, 0), j=0..n/i)))
    end:
a:= n-> g(2*n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 06 2015

g[n_, i_, p_] := g[n, i, p] = If[n == 0, p!, If[i<1, 0, Sum[If[Mod[j, 2] == 0, g[n - i*j, i-1, p + j*i]/j!/i!^j, 0], {j, 0, n/i}]]]; a[n_] := g[2*n, 2*n, 0]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, May 12 2015, after Alois P. Heinz *)

E.g.f.: Product_{k>0} cosh(x^k/k!).

More terms from Emeric Deutsch, Sep 01 2007

A352141 Numbers whose prime factorization has all even indices and all even exponents.

Original entry on oeis.org

1, 9, 49, 81, 169, 361, 441, 729, 841, 1369, 1521, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 6241, 6561, 7569, 7921, 8281, 10201, 11449, 12321, 12769, 13689, 16641, 17161, 17689, 19321, 21609, 22801, 25281, 26569, 28561, 29241, 29929, 32761, 33489, 35721

Offset: 1

Gus Wiseman, Mar 18 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of partitions with all even parts and all even multiplicities, counted by A035444.

			The terms together with their prime indices begin:
     1 = 1
     9 = prime(2)^2
    49 = prime(4)^2
    81 = prime(2)^4
   169 = prime(6)^2
   361 = prime(8)^2
   441 = prime(2)^2 prime(4)^2
   729 = prime(2)^6
   841 = prime(10)^2
  1369 = prime(12)^2
  1521 = prime(2)^2 prime(6)^2
  1849 = prime(14)^2
  2401 = prime(4)^4
  2809 = prime(16)^2
  3249 = prime(2)^2 prime(8)^2
  3721 = prime(18)^2
  3969 = prime(2)^4 prime(4)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The second condition alone (all even exponents) is A000290, counted by A035363.
The restriction to primes is A031215.
These partitions are counted by A035444.
The first condition alone is A066207, counted by A035363, squarefree A258117.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.

Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from itertools import count, islice
from sympy import factorint, primepi
def A352141_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())),count(max(startvalue,1)))
A352141_list = list(islice(A352141_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A000290 and A066207.
A257991(a(n)) = A162642(a(n)) = 0.
A257992(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022

A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.

Original entry on oeis.org

1, 1, 4, 86, 2696, 168232, 15948032, 2172623168, 398846422144, 97541017510784, 29909993927387648, 11447388459863715328, 5284740632299379566592, 2927671399386587378671616, 1897593132067741963020476416, 1437515129453860805943287939072

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(2)=4 because we have (1)(2)(3)(4), (12)(34), (13)(24) and (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..220

Cf. A055922, A130219, A130220, A130263, A218504, A270597, A270598.

g:=product(cosh(x^k/k),k=1..30): gser:=series(g,x=0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=0, multinomial(n, n-i*j, i$j)
       *(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

nn=26;Select[Range[0,nn]!CoefficientList[Series[Product[Cosh[x^k/k],{k,1,nn}],{x,0,nn}],x],#>0&] (* Geoffrey Critzer, Sep 17 2013 *)

E.g.f.: Product_{k>0} cosh(x^k/k).

a(n) ~ c * (2*n-1)! / n ~ c * sqrt(Pi) * n^(2*n-3/2) * 2^(2*n) / exp(2*n), where c = A249673 = Product_{k>=1} cosh(1/k) = 2.1164655365... . - Vaclav Kotesovec, Mar 19 2016

More terms from Emeric Deutsch, Aug 24 2007

A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.

Original entry on oeis.org

1, 3, 7, 13, 19, 21, 27, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317

Offset: 1

Gus Wiseman, Mar 11 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.
All terms are odd. - Michael S. Branicky, Mar 12 2022

			The terms together with their prime indices begin:
      1 = 1
      3 = prime(2)^1
      7 = prime(4)^1
     13 = prime(6)^1
     19 = prime(8)^1
     21 = prime(4)^1 prime(2)^1
     27 = prime(2)^3
     29 = prime(10)^1
     37 = prime(12)^1
     39 = prime(6)^1 prime(2)^1
     43 = prime(14)^1
     53 = prime(16)^1
     57 = prime(8)^1 prime(2)^1
     61 = prime(18)^1
     71 = prime(20)^1

The restriction to primes is A031215.
These partitions are counted by A055922 (aerated).
The first condition alone is A066207, counted by A035363.
The squarefree case is A258117.
The second condition alone is A268335, counted by A055922.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.

Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    if n%2 == 0: return False
    return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
print([k for k in range(318) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Intersection of A066207 and A268335.
A257991(a(n)) = A162641(a(n)) = 0.
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A001222(a(n)).

A130220 Number of partitions of n-set in which number of blocks of size k is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 5, 5, 27, 117, 331, 1213, 6579, 47193, 140527, 1213841, 4617927, 48210879, 243443739, 2392565149, 10377087115, 125434781845, 725455816883, 8086277450629, 59694530600595, 614469256831895, 4650128350629285, 52385811781286769, 467607504075929863

Offset: 0

Vladeta Jovovic, Aug 04 2007, Aug 05 2007

			a(4)=5 because we have abcd, a|bcd, acd|b, abd|c and abc|d.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A055922, A102759, A111723, A111724.

g:=product(1+sinh(x^k/factorial(k)),k=1..30): gser:=series(g,x=0,28): seq(factorial(n)*coeff(gser,x,n),n=0..24); # Emeric Deutsch, Sep 01 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)
       /j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[j == 0 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

E.g.f.: Product_{k>0} (1+sinh(x^k/k!)).

More terms from Emeric Deutsch, Sep 01 2007

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

Offset: 1

Gus Wiseman, Mar 11 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

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

cp's OEIS Frontend

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