A037032 -id:A037032 - cp's OEIS frontend

A073118 Total sum of prime parts in all partitions of n.

0, 2, 5, 9, 19, 33, 57, 87, 136, 206, 311, 446, 650, 914, 1284, 1762, 2432, 3276, 4433, 5888, 7824, 10272, 13479, 17471, 22642, 29087, 37283, 47453, 60306, 76112, 95931, 120201, 150338, 187141, 232507, 287591, 355143, 436849, 536347, 656282, 801647, 976095

Offset: 1

Vladeta Jovovic, Aug 24 2002

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                          Sum of
Partitions              prime parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 2
2 + 2 + 2 .................. 6
5 + 1 ...................... 5
3 + 2 + 1 .................. 5
4 + 1 + 1 .................. 0
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 2
1 + 1 + 1 + 1 + 1 + 1 ...... 0
--------------------------------------
Total ..................... 33
So a(6) = 33. (End)

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

Cf. A037032, A194545, A309561.

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(isprime(i), t[1]*i*j, 0)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2011

f[n_] := Apply[Plus, Select[ Flatten[ IntegerPartitions[n]], PrimeQ[ # ] & ]]; Table[ f[n], {n, 1, 41} ]
a[n_] := Sum[Total[FactorInteger[k][[All, 1]]]*PartitionsP[n-k], {k, 1, n}] - PartitionsP[n-1]; Array[a, 50] (* Jean-François Alcover, Dec 27 2015 *)

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

a(n) = Sum_{k=1..n} A008472(k)*A000041(n-k).

G.f.: Sum_{i>=1} prime(i)*x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Feb 01 2017

Edited and extended by Robert G. Wilson v, Aug 26 2002

A102291 Total number of prime parts in all compositions of n.

Original entry on oeis.org

0, 0, 1, 3, 7, 18, 42, 98, 222, 497, 1100, 2413, 5250, 11350, 24398, 52193, 111180, 235949, 499074, 1052502, 2213710, 4644833, 9724492, 20318637, 42376578, 88231765, 183420748, 380755932, 789340736, 1634339217, 3379993922, 6982618822, 14410499598, 29711523105

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..3313

Cf. A037032, A224344, A336632.

a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 06 2019

a[n_] := a[n] = If[n==0, 0, Sum[a[n-j] + If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}]];
a /@ Range[0, 33] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

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

a(n) = Sum_{k=1..floor(n/2)} k * A224344(n,k). - Alois P. Heinz, Aug 06 2019

More terms from Joshua Zucker, May 10 2006

A144115 Total number of Fibonacci parts in all partitions of n.

Original entry on oeis.org

1, 3, 6, 11, 19, 32, 49, 77, 114, 169, 241, 345, 480, 667, 910, 1237, 1656, 2213, 2918, 3840, 5003, 6497, 8368, 10751, 13711, 17441, 22052, 27806, 34879, 43645, 54355, 67535, 83571, 103171, 126907, 155766, 190554, 232629, 283158, 343969, 416716, 503900, 607807

Offset: 1

Omar E. Pol, Sep 11 2008

nonn

a(n) is also the sum of the differences between the sum of f-th largest and the sum of (f+1)-st largest elements in all partitions of n for all Fibonacci parts f. - Omar E. Pol, Oct 27 2012

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                        Number of
Partitions            Fibonacci parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
5 + 1 ...................... 2
3 + 2 + 1 .................. 3
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 32
So a(6) = 32. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000045, A006128, A037032, A144116, A144117, A144118, A199936, A309537.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4)
      )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60); # Alois P. Heinz, Jun 24 2009, revised Aug 06 2019

Clear[b]; b[] = False; l = {0, 1}; For[k=1, k <= 100, k++, b[l[[1]]] = True; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n, i_] := aa[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i==0 || n<0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]]; a[n_] := aa[n, n][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)

G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017

More terms from Alois P. Heinz, Jun 24 2009

A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 3, 4, 3, 1, 3, 6, 4, 2, 5, 7, 6, 3, 1, 6, 9, 8, 5, 2, 8, 11, 12, 7, 3, 1, 8, 17, 14, 10, 5, 2, 12, 20, 19, 14, 8, 3, 1, 13, 26, 25, 19, 11, 5, 2, 17, 31, 35, 24, 16, 8, 3, 1, 19, 41, 42, 34, 21, 12, 5, 2, 26, 47, 56, 44, 29

Offset: 0

Alois P. Heinz, May 29 2013

			T(6,0) = 3: [6], [4,1,1], [1,1,1,1,1,1].
T(6,1) = 4: [5,1], [4,2], [3,1,1,1], [2,1,1,1,1].
T(6,2) = 3: [3,3], [3,2,1], [2,2,1,1].
T(6,3) = 1: [2,2,2].
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2;
  2,  2,  1;
  2,  3,  2;
  3,  4,  3,  1;
  3,  6,  4,  2;
  5,  7,  6,  3, 1;
  6,  9,  8,  5, 2;
  8, 11, 12,  7, 3, 1;
  8, 17, 14, 10, 5, 2;
  ...

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

Column k=0 gives: A002095.
Row sums give: A000041.
Cf. A000040, A004526, A224344.

b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1 then 0
      else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(isprime(i), 0$j, NULL), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n$2):
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]]]]; 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]]; T[n_] := b[n, n]; 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) = A037032(n).

G.f.: G(t,x) = Product_{i>=1} (1 - x^prime(i))/((1 - x^i)*(1 - t*x^prime(i))). - Emeric Deutsch, Nov 11 2015

A073335 Total number of prime power parts in all partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 15, 23, 39, 58, 89, 128, 189, 264, 375, 515, 713, 960, 1301, 1726, 2298, 3011, 3948, 5113, 6625, 8492, 10880, 13825, 17545, 22108, 27823, 34800, 43465, 54003, 66983, 82709, 101960, 125180, 153432, 187397, 228490, 277707, 336972

Offset: 1

Vladeta Jovovic, Aug 22 2002

			a(4)=5 because in all partitions of 4 we have 5 powers of primes (shown between parentheses): (4), (3)1, (2)(2), (2)11, 1111.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000041, A001222, A037032.

with(numtheory): with(combinat): a:= n-> add(bigomega(k)*numbpart(n-k), k=1..n): seq(a(n), n=1..46); # Emeric Deutsch, Feb 26 2005

Table[Sum[PrimeOmega[k]*PartitionsP[n - k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, May 05 2017 *)

a(n) = sum(k=1, n, bigomega(k)*numbpart(n-k)); \\ Michel Marcus, May 05 2017

a(n) = Sum_{k=1..n} bigomega(k)*numbpart(n-k).

G.f.: Sum_{i>=2} floor(1/omega(i))*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j), where omega() is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 24 2017

More terms from Emeric Deutsch, Feb 26 2005

A144119 Total number of nonprime parts in all partitions of n.

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 34, 54, 80, 119, 170, 246, 342, 478, 653, 894, 1198, 1610, 2127, 2813, 3672, 4789, 6181, 7975, 10192, 13010, 16488, 20861, 26224, 32918, 41086, 51199, 63494, 78599, 96888, 119235, 146167, 178879, 218181, 265662, 322487, 390834, 472343

Offset: 1

Omar E. Pol, Sep 11 2008

a(n) is also the sum of the differences between the sum of m-th largest and the sum of (m+1)st largest elements in all partitions of n for all nonprimes m. - Omar E. Pol, Oct 27 2012

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                        Number of
Partitions            nonprime parts
--------------------------------------
6 .......................... 1
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 3
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 22
So a(6) = 22. (End)

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

Cf. A006128, A018252, A037032, A144116, A144121.

b:= proc(n, i) option remember; local g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(isprime(i), 0, g[1])]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

b[n_, i_] := b[n, i] = Module[{g}, If[n == 0, {1, 0}, If[i<1, {0, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[PrimeQ[i], 0, g[[1]]]} ]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ Altug Alkan, Oct 29 2015

a(n) = A006128(n)-A037032(n).

A144120 Number of prime parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 32, 48, 61, 88, 113, 154, 198, 267, 337, 446, 563, 730, 915, 1174, 1460, 1853, 2294, 2878, 3545, 4416, 5404, 6679, 8144, 9991, 12125, 14791, 17866, 21677, 26084, 31478, 37733, 45340

Offset: 1

Omar E. Pol, Sep 11 2008

First differences of A037032.

Cf. A037032, A135010, A138121, A138137, A144117, A144118, A144121.

a(n) = A037032(n)-A037032(n-1).

A183088 Total number of parts that are partition numbers A000041 in all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 50, 77, 115, 170, 244, 348, 485, 674, 922, 1251, 1678, 2241, 2959, 3892, 5076, 6592, 8497, 10915, 13930, 17719, 22417, 28267, 35474, 44395, 55312, 68730, 85082, 105049, 129261, 158675, 194171, 237077, 288651

Offset: 0

Omar E. Pol, Aug 05 2011

			a(5) = 19 because the 7 partitions of 5 are [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and we can see that there are 19 parts that are partition numbers A000041. Note that there are 20 parts but the 4 is not a partition number, so a(5) = 20 - 1 = 19.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A037032, A144115.

A000041 = Table[PartitionsP[n], {n, 0, 45}]; Table[Length[Select[Flatten[IntegerPartitions[n]], MemberQ[A000041, #] &]], {n, 40}] (* Alonso del Arte, Aug 05 2011 *)

A281573 Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 6, 11, 19, 33, 51, 79, 118, 176, 252, 362, 505, 705, 965, 1314, 1765, 2365, 3127, 4124, 5387, 7012, 9052, 11653, 14893, 18982, 24048, 30378, 38176, 47857, 59704, 74302, 92099, 113879, 140300, 172463, 211297, 258325, 314887, 383037, 464684, 562653, 679566, 819269, 985449, 1183242, 1417738, 1695886

Offset: 1

Ilya Gutkovskiy, Jan 24 2017

nonn

Total number of squarefree parts in all partitions of n.

Convolution of A000041 and A034444.

			a(5) = 19 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 3 + 3 + 4 + 5 = 19.

Index entries for related partition-counting sequences

Cf. A000041, A005117, A008683, A034444, A037032, A073335, A073336.

nmax = 48; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i), {i, 1, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

A326957 Total number of noncomposite parts in all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 50, 77, 115, 170, 244, 348, 486, 675, 923, 1253, 1682, 2246, 2968, 3904, 5094, 6616, 8533, 10962, 13997, 17808, 22538, 28426, 35689, 44670, 55678, 69199, 85692, 105826, 130261, 159935, 195778, 239092, 291191, 353854, 428925, 518848

Offset: 0

Omar E. Pol, Aug 08 2019

nonn

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             noncomposite
of 6                       parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
5 + 1 ...................... 2
3 + 2 + 1 .................. 3
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 32
So a(6) = 32.

First differs from A183088 at a(13).

Cf. A000041, A000070, A006128, A008578 (noncomposites), A037032, A144115, A144116, A144119, A326958, A326981.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), p[1], 0)])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019

b[n_] := Sum[PrimeNu[k] PartitionsP[n-k], {k, 1, n}];
c[n_] := SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}]/(1-x), {x, 0, n}];
a[n_] := b[n] + c[n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 15 2020 *)

a(n) = A037032(n) + A000070(n-1), n >= 1.

a(n) = A006128(n) - A326981(n).

cp's OEIS Frontend

A073118 Total sum of prime parts in all partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A102291 Total number of prime parts in all compositions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A144115 Total number of Fibonacci parts in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A073335 Total number of prime power parts in all partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A144119 Total number of nonprime parts in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A144120 Number of prime parts in the last section of the set of partitions of n.

Views

Author

Keywords

Comments