A222656 -id:A222656 - cp's OEIS frontend

A002095 Number of partitions of n into nonprime parts.

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 13, 17, 19, 26, 28, 37, 40, 52, 58, 73, 79, 102, 113, 139, 154, 191, 210, 258, 284, 345, 384, 462, 509, 614, 679, 805, 893, 1060, 1171, 1382, 1528, 1792, 1988, 2319, 2560, 2986, 3304, 3823, 4231, 4888, 5399, 6219, 6870

Offset: 0

N. J. A. Sloane

Partial sums of A023895. - Emeric Deutsch, Apr 19 2006

Column k=0 of A222656. - Alois P. Heinz, May 29 2013

			a(6) = 3 from the partitions 6 = 1+1+1+1+1+1 = 4+1+1.

L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
A. Murthy, Some new Smarandache sequences, functions and partitions, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000 (but beware errors).
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 2.6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from T. D. Noe)

Cf. A000607, A018252, A096258.

a002095 = p a018252_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 15 2012

g:=product((1-x^ithprime(j))/(1-x^j),j=1..60): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Apr 19 2006

NonPrime[n_Integer] := FixedPoint[n + PrimePi[ # ] &, n + PrimePi[n]]; CoefficientList[ Series[1/Product[1 - x^NonPrime[i], {i, 1, 50}], {x, 0, 50}], x]

first(n)=my(x='x+O('x^(n+1)),pr=1); forprime(p=2,n+1, pr *= (1-x^p)); pr/prod(i=1,n+1, 1-x^i) \\ Charles R Greathouse IV, Jun 23 2017

G.f.: Product_{i>0} (1-x^prime(i))/(1-x^i). - Vladeta Jovovic, Jul 31 2004

More terms from James Sellers, Dec 23 1999

Corrected by Robert G. Wilson v, Feb 11 2002

A037032 Total number of prime parts in all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 20, 32, 48, 73, 105, 153, 214, 302, 415, 569, 767, 1034, 1371, 1817, 2380, 3110, 4025, 5199, 6659, 8512, 10806, 13684, 17229, 21645, 27049, 33728, 41872, 51863, 63988, 78779, 96645, 118322, 144406, 175884, 213617, 258957, 313094, 377867

Offset: 1

G. L. Honaker, Jr.

nonn

a(n) is also the sum of the differences between the sum of p-th largest and the sum of (p+1)st largest elements in all partitions of n for all primes p. - Omar E. Pol, Oct 25 2012

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                        Number of
Partitions              prime parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
5 + 1 ...................... 1
3 + 2 + 1 .................. 2
4 + 1 + 1 .................. 0
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 1
2 + 1 + 1 + 1 + 1 .......... 1
1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
Total ..................... 13
So a(6) = 13.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A001221, A073118.

with(combinat): a:=proc(n) local P,c,j,i: P:=partition(n): c:=0: for j from 1 to numbpart(n) do for i from 1 to nops(P[j]) do if isprime(P[j][i])=true then c:=c+1 else c:=c fi: od: od: c: end: seq(a(n),n=1..42); # Emeric Deutsch, Mar 30 2006
# second Maple program
b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(isprime(i), g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

a[n_] := Sum[PrimeNu[k]*PartitionsP[n - k], {k, 1, n}]; Array[a, 100] (* Jean-François Alcover, Mar 16 2015, after Vladeta Jovovic *)

a(n)={sum(k=1, n, omega(k)*numbpart(n-k))} \\ Andrew Howroyd, Dec 28 2017

a(n) = Sum_{k=1..n} A001221(k)*A000041(n-k). - Vladeta Jovovic, Aug 22 2002
a(n) = Sum_{k=1..floor(n/2)} k*A222656(n,k). - Alois P. Heinz, May 29 2013
G.f.: Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017

More terms from Naohiro Nomoto, Apr 19 2002

A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1

Offset: 0

Alois P. Heinz, May 23 2013

			A(5,1) = 8: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [3,1,1], [1,3,1], [1,1,3], [5].
Triangle T(n,k) begins:
   1;
   1;
   1,   1;
   1,   3;
   2,   5,   1;
   3,   8,   5;
   5,  13,  13,   1;
   7,  23,  27,   7;
  11,  39,  52,  25,   1;
  17,  65,  99,  66,   9;
  27, 106, 186, 151,  41,  1;
  40, 177, 340, 323, 133, 11;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives: A052284.
Row sums are: A011782.
Row lengths are: A008619.
T(floor(n/2)) = A093178(n).
T(2n,n-1) = A001844(n-1) for n>0.
Cf. A000040, A000079, A004526, A102291, A121303, A222656.

T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)->x+y, %,
      [`if`(isprime(j), 0, NULL), T(n-j)], 0) od; %[] fi
    end:
seq(T(n), n=0..16);

zip[f_, x_List, y_List, z_] :=  With[{m = Max[Length[x], Length[y]]},  Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; T[n_] := T[n] =  Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= n, j++, pc = zip[Plus, pc, Join[If[PrimeQ[j], {0}, {}], T[n-j]], 0]]; pc]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Sum_{k=1..floor(n/2)} k * T(n,k) = A102291(n).

A299730 Irregular triangle read by rows: T(n,k) is the number of partitions of 3n having exactly k prime parts; n >= 0, 0 <= k <= floor( 3n / 2 ).

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 1, 6, 9, 8, 5, 2, 12, 20, 19, 14, 8, 3, 1, 19, 41, 42, 34, 21, 12, 5, 2, 37, 72, 88, 74, 53, 31, 18, 8, 3, 1, 58, 136, 161, 155, 115, 77, 46, 25, 12, 5, 2, 102, 226, 307, 291, 241, 168, 110, 65, 35, 18, 8, 3, 1

Offset: 0

J. Stauduhar, Feb 17 2018

Sequence of row lengths = A001651.

			The irregular triangle T(n, k) begins:
3n\k  0   1   2   3   4   5   6   7   8   9
0:    1
3:    1   2
6:    3   4   3   1
9:    6   9   8   5   2
12:  12  20  19  14   8   3   1
15:  19  41  42  34  21  12   5   2
18:  37  72  88  74  53  31  18   8   3   1

J. Stauduhar, Table of n, a(n) for n = 0..719

Cf. A001651, A008585, A222656.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)*`if`(isprime(i), x^j, 1), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(3*n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 03 2018

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]*If[PrimeQ[i], x^j, 1], {j, 0, n/i}]]];
T[n_] := CoefficientList[b[3n, 3n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Mar 08 2021, after Alois P. Heinz *)

T(n,k) = A222656(3n,k).

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).

A299731 Number of partitions of 3*n that have exactly n prime parts.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843, 5960, 7312, 8957, 10925, 13291, 16139, 19534, 23588, 28437, 34180, 41000, 49099, 58657, 69941, 83269, 98917, 117314, 138930

Offset: 0

J. Stauduhar, Feb 18 2018

nonn

			For n = 3: the five partitions of 3 * 3 = 9 that have exactly three prime parts are (5, 2, 2), (3, 3, 3), (3, 3, 2, 1), (3, 2, 2, 1, 1), and (2, 2, 2, 1, 1, 1), so a(3) = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
J. Kelleher, B. O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009, 2014.
J. Stauduhar, Python program

Cf. A222656, A299730, A299732.

zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[ PadRight[x, m, z], PadRight[y, m, z]]]];
b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1}, i < 1, {0}, True, pc = {}; For[j = 0, j <= n/i, j++, pc = zip[Plus, pc, Join[If[PrimeQ[i], Array[0 &, j], {}], b[n - i*j, i - 1]], 0]]; pc]];
a[n_] := b[3 n, 3 n][[n + 1]];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 16 2018, after Alois P. Heinz *)

a(n) = {my(nb = 0); forpart(p=3*n, if (sum(k=1, #p, isprime(p[k])) == n, nb++);); nb;} \\ Michel Marcus, Mar 22 2018

Python
```
See Stauduhar link.
```

a(n) = A222656(3*n,n).

A299732 a(n) has exactly (a(n) - n) / 2 partitions with exactly (a(n) - n) / 2 prime parts.

Original entry on oeis.org

2, 5, 8, 13, 20, 29, 42, 57, 78, 109, 148, 197, 264, 347, 454, 595, 770, 989, 1272, 1619, 2054, 2601, 3268, 4087, 5108, 6347, 7860, 9713, 11948, 14653, 17944, 21881, 26614, 32311, 39102, 47211, 56910, 68397, 82038, 98237, 117354, 139923, 166580, 197877, 234672

Offset: 0

J. Stauduhar, Feb 18 2018

nonn

If B={b(n)} is the complement of A299731 then no number exists that has exactly b(n) partitions that have exactly b(n) prime parts, so this sequence lists only those numbers that can have the equality property.

Up to a(44) = 234672 (currently, the last term), except for 2,5,8, and 29, every term is the sum of distinct previous terms. Will this be true for all new terms?

			For n = 3: A299731(3) = 5. a(3) = 2*5 + 3 = 13. The five partitions of 13 that have exactly five prime parts are: (5,2,2,2,2), (3,3,3,2,2), (3,3,2,2,2,1), (3,2,2,2,2,1,1), and (2,2,2,2,2,1,1,1), so a(3) = 13.

J. Stauduhar, Python program.

Cf. A222656, A299730, A299731.

Python
```
# See Stauduhar link.
```

a(n) = 2*A299731(n) + n = 2*A222656(3*n,n) + n.

cp's OEIS Frontend

A002095 Number of partitions of n into nonprime parts.

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A037032 Total number of prime parts in all partitions of n.

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A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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A299730 Irregular triangle read by rows: T(n,k) is the number of partitions of 3*n having exactly k prime parts; n >= 0, 0 <= k <= floor( 3*n / 2 ).

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

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A299731 Number of partitions of 3*n that have exactly n prime parts.

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A299732 a(n) has exactly (a(n) - n) / 2 partitions with exactly (a(n) - n) / 2 prime parts.

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A299730 Irregular triangle read by rows: T(n,k) is the number of partitions of 3n having exactly k prime parts; n >= 0, 0 <= k <= floor( 3n / 2 ).