A002095 -id:A002095 - cp's OEIS frontend

A355306 Number of partitions of n in which the number of prime parts is not equal to the number of nonprime parts.

0, 1, 2, 2, 4, 7, 8, 13, 19, 25, 38, 48, 65, 91, 120, 153, 209, 264, 343, 443, 563, 713, 912, 1133, 1428, 1789, 2217, 2746, 3406, 4178, 5139, 6296, 7670, 9344, 11360, 13732, 16612, 20038, 24078, 28915, 34660, 41402, 49439, 58887, 69983, 83088, 98476, 116436, 137589, 162244, 191018

Offset: 0

Omar E. Pol, Jun 28 2022

nonn

			For n = 6 the partitions of 6 in which the number of prime parts is not equal to the number of nonprime parts are [6], [3, 3], [2, 2, 2], [3, 2, 1], [4, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], there are eight of these partitions so a(6) = 8.

Index entries for sequences related to partitions

Cf. A000040, A000041, A000607, A002095, A002096, A018252, A155515, A355158, A355225.

Array[Count[IntegerPartitions[#], ?(#1 - #2 != #2 & @@ {Length[#], Count[#, ?PrimeQ]} &)] &, 51, 0] (* Michael De Vlieger, Jul 15 2022 *)

a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) != #p/2, nb++)); nb; \\ Michel Marcus, Jun 30 2022

from sympy import isprime
from sympy.utilities.iterables import partitions
def c(p): return 2*sum(p[i] for i in p if isprime(i)) != sum(p.values())
def a(n): return sum(1 for p in partitions(n) if c(p))
print([a(n) for n in range(51)]) # Michael S. Branicky, Jun 28 2022

a(n) = A000041(n) - A155515(n).

a(n) = A355158(n) + A355225(n).

A132381 Number of partitions of n with exactly one prime number.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 9, 12, 15, 22, 28, 38, 46, 62, 77, 98, 117, 152, 183, 230, 275, 344, 408, 504, 592, 726, 856, 1038, 1212, 1469, 1712, 2048, 2380, 2839, 3288, 3901, 4500, 5313, 6127, 7193, 8254, 9671, 11081, 12909, 14764, 17153, 19566, 22658, 25786, 29762

Offset: 1

Reinhard Zumkeller, Nov 10 2007

nonn

			a(10) = #{8+2, 7+1+1+1, 6+3+1, 6+2+2, 6+2+1+1, 5+5, 5+4+1, 5+1+1+1+1+1, 4+4+2, 4+3+3, 4+3+1+1+1, 4+2+2+2, 4+2+2+1+1, 4+2+1+1+1+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1, 2+2+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1} = 22.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A002095.

Column k=1 of A274517.

b:= proc(n, i) option remember; local j; if n=0 then [1, 0] elif i<1
      then [0$2] else b(n, i-1); for j to n/i do zip((x, y)->x+y, %,
      [`if`(isprime(i), 0, NULL), b(n-i*j, i-1)[]], 0) od; %[1..2] fi
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, May 29 2013

zip = With[{m = Max[Length[#1], Length[#2]]}, PadRight[#1, m] + PadRight[#2, m]]&; b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1, 0}, i<1, {0, 0}, True, pc = b[n, i-1]; For[j = 1, j <= n/i, j++, pc = zip[pc, Join[{If[PrimeQ[i], 0, Nothing]}, b[n-i*j, i-1]]] ]; pc[[1 ;; 2]] ]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 12 2017, after Alois P. Heinz *)

A236019 Smallest number having at least n partitions that contain at least n primes.

Original entry on oeis.org

0, 2, 5, 8, 10, 13, 15, 17, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130

Offset: 0

J. Stauduhar, Jan 18 2014

nonn

			a(4) = 10: [2,2,2,2,1,1], [2,2,2,2,2], [3,2,2,2,1], [3,3,2,2].

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

Cf. A000041, A002095, A235945, A236444 (complement).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, t) +`if`(i>n, 0,
      b(n-i, i, t -`if`(t>0 and isprime(i), 1, 0)))))
    end:
a:= proc(n) option remember; local k;
      for k from a(n-1) while b(k, k, n)Alois P. Heinz, Jan 18 2014

$RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t - If[t>0 && PrimeQ[i], 1, 0]]]]]; a[n_] := a[n] = Module[{k}, For[k = a[n-1], b[k, k, n] < n, k++]; k]; a[0] = 0; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jan 18 2014

A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 1, 3, 7, 1, 3, 9, 3, 5, 12, 5, 6, 15, 9, 8, 22, 11, 1, 8, 28, 19, 1, 12, 38, 24, 3, 13, 46, 38, 4, 17, 62, 48, 8, 19, 77, 68, 12, 26, 98, 87, 20, 28, 117, 127, 24, 1, 37, 152, 154, 41, 1, 40, 183, 210, 55, 2, 52, 230, 260, 82, 3

Offset: 0

Geoffrey Critzer, Jun 25 2016

Row lengths increase by 1 at row A007504(n).
Columns k=0-1 give: A002095, A132381.
Row sums give: A000041.

			T(6,1) = 7 because we have: 5+1, 4+2, 3+3, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1+1.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2;
  2,  3;
  2,  4,  1;
  3,  7,  1;
  3,  9,  3;
  5, 12,  5;
  6, 15,  9;
  8, 22, 11, 1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A002095, A007504, A132381, A222656.

b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j>0 and isprime(i), x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Jun 26 2016

nn = 20; Map[Select[#, # > 0 &] &, CoefficientList[Series[Product[
      1/(1 - z^k), {k,Select[Range[1000], PrimeQ[#] == False &]}] Product[
      u/(1 - z^j) - u + 1, {j, Table[Prime[n], {n, 1, nn}]}], {z, 0,
     nn}], {z, u}]] // Grid

G.f.: Product_{k>=1} (1 - x^prime(k))/(1 - x^k)*(y/(1-x^prime(k)) - y + 1).

A280285 Number of partitions of n into odd composite numbers (A071904).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 3, 0, 0, 3, 1, 0, 4, 1, 1, 5, 1, 0, 5, 2, 2, 6, 2, 1, 8, 3, 1, 8, 3, 2, 11, 3, 2, 12, 5, 4, 13, 5, 3, 16, 8, 4, 18, 7, 6, 22, 9, 7, 24, 12, 9, 28, 12, 9, 33, 18, 11, 36, 18, 14, 45, 22, 16, 48, 26, 22, 54, 29, 23, 66, 38

Offset: 0

Ilya Gutkovskiy, Dec 31 2016

nonn

			a(36) = 3 because we have [27, 9], [21, 15] and [9, 9, 9, 9].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Composite Number
Index entries for related partition-counting sequences

Cf. A002095, A002808, A023895, A071904, A204389, A280287.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d>1 and d::odd and not isprime(d), d, 0),
       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 31 2016

nmax = 100; CoefficientList[Series[(1 - x)/(1 - x^2) Product[(1 - x^(2 k)) (1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: ((1 - x)/(1 - x^2))*Product_{k>=1} (1 - x^(2*k))*(1 - x^prime(k))/(1 - x^k).

A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, -1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1, -1, 1, -2, 1, -1, 0, 1, -1, 1, -2, 2, -1, -1, 2, -1, -1, 2, -2, 2, -1, 1, 0, -1, 1, 0, 1, -2, 2, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, -2, 0, -1, 0, 0, -2, 2, -3, 0, 2, -2, 1, -1, 1, -2, 1, -1, -1, 1

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

sign

The difference between the number of partitions of n into an even number of distinct nonprime parts and the number of partitions of n into an odd number of distinct nonprime parts.

Convolution of the sequences A000607 and A010815.

Index entries for sequences related to partitions

Cf. A000607, A002095, A010815, A018252, A046675, A096258, A302236.

nmax = 80; CoefficientList[Series[Product[(1 - x^k)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[(1 - Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A018252(k)).

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23

Offset: 1

Gus Wiseman, Dec 14 2018

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79

Offset: 1

Gus Wiseman, Dec 14 2018

			48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).

Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.

nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A341460 Number of partitions of n into 10 nonprime parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 60, 68, 79, 92, 104, 122, 139, 157, 181, 208, 234, 270, 304, 347, 391, 445, 499, 569, 636, 724, 805, 913, 1015, 1150, 1274, 1440, 1592, 1796, 1980, 2231, 2455

Offset: 10

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259201, A341408, A341451, A341452, A341453, A341454, A341455, A341457.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i), t-1))))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=10..69);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 10, 69}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{10}],?(NoneTrue[#,PrimeQ]&)],{n,10,70}] (* _Harvey P. Dale, Sep 01 2024 *)

A347663 Number of partitions of n into at most 5 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 5, 8, 8, 10, 10, 15, 14, 20, 19, 25, 25, 33, 30, 43, 42, 53, 52, 68, 64, 84, 80, 102, 100, 126, 119, 154, 148, 181, 177, 219, 209, 261, 251, 304, 299, 359, 344, 420, 408, 484, 475, 564, 546, 648, 632, 739, 728, 849, 825, 968

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Cf. A002095, A018252, A307857, A341452, A347662, A347664.

cp's OEIS Frontend

A355306 Number of partitions of n in which the number of prime parts is not equal to the number of nonprime parts.

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A132381 Number of partitions of n with exactly one prime number.

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A236019 Smallest number having at least n partitions that contain at least n primes.

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A274517 Number T(n,k) of integer partitions of n with exactly k distinct primes.

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A280285 Number of partitions of n into odd composite numbers (A071904).

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Maple

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A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

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A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

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A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

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