A258323 -id:A258323 - cp's OEIS frontend

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

Offset: 0

Alois P. Heinz, Sep 05 2014

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

			The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3;
   5,  6;
   7, 10;
  11, 14, 15;
  13, 21, 22, 30;
  17, 26, 33, 35, 42;
  19, 34, 39, 55, 66,  70;
  23, 38, 51, 65, 77,  78, 105, 110;
  29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
  ...
Corresponding triangle of strict integer partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (42) (51) (321)
        (7) (61) (52) (43) (421)
     (8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018

Alois P. Heinz, Rows n = 0..42, flattened

Column k=1 gives: A008578(n+1).
Last elements of rows give: A246868.
Row sums give A147655.
Row lengths are: A000009.
Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
      map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

Original entry on oeis.org

1, 2, 3, 11, 17, 40, 86, 153, 283, 547, 1069, 1737, 3238, 5340, 9574, 17251, 27897, 45845, 78601, 126725, 207153, 353435, 550422, 881454, 1393870, 2239938, 3473133, 5546789, 8762663, 13341967, 20676253, 31774563, 48248485, 74174759, 111904363, 170184798

Offset: 0

Neil Fernandez, Nov 09 2008

nonn

Sum of all squarefree numbers whose prime indices sum to n. 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. - Gus Wiseman, May 09 2019

			Form a product from the primes: (1 + 2*x) * (1 + 3*x^2) * (1 + 5*x^3) * ...* (1 + prime(n)*x^n) * ... Multiplying out gives 1 + 2*x + 3*x^2 + 11*x^3 + ..., so the sequence begins 1, 2, 3, 11, ....
From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get:
a(1) = f(1) = 2,
a(2) = f(2) = 3,
a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,
a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 13 + 2*11 + 3*7 + 2*3*5 = 86,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 17 + 2*13 + 3*11 + 5*7 + 2*3*7 = 153. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9000 (first 1001 terms from Harvey P. Dale)

Row sums of A246867 and A258323.

Cf. A000009,A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 05 2014

nn=40;Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n,{n,nn}]],x]],nn] (* Harvey P. Dale, Jul 01 2012 *)

a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - Alois P. Heinz, Sep 05 2014

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = prime(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 10 2020

More terms from Harvey P. Dale, Jul 01 2012
a(0)=1 inserted by Alois P. Heinz, Sep 05 2014
Name edited by Petros Hadjicostas, Apr 10 2020

A025129 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.

Original entry on oeis.org

0, 6, 10, 29, 43, 94, 128, 231, 279, 484, 584, 903, 1051, 1552, 1796, 2489, 2823, 3784, 4172, 5515, 6091, 7758, 8404, 10575, 11395, 14076, 15174, 18339, 19667, 23414, 24906, 29437, 31089, 36500, 38614, 44731, 47071, 54198, 56914, 65051, 68371, 77402, 81052, 91341

Offset: 1

Clark Kimberling

nonn

This is the sum of distinct squarefree semiprimes with prime indices summing to n + 1. A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Dec 05 2020

			From _Gus Wiseman_, Dec 05 2020: (Start)
The sequence of sums begins (n > 1):
    6 =  6
   10 = 10
   29 = 14 + 15
   43 = 22 + 21
   94 = 26 + 33 + 35
  128 = 34 + 39 + 55
  231 = 38 + 51 + 65 + 77
  279 = 46 + 57 + 85 + 91
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sum of prime(i) * prime(j) for i + j = n, i != j.

Cf. A000040, A258323.
The nonsquarefree version is A024697 (shifted right).
Row sums of A338905 (shifted right).
A332765 is the greatest among these squarefree semiprimes.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A014342 is the self-convolution of the primes.
A056239 is the sum of prime indices of n.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339194 sums squarefree semiprimes grouped by greater prime factor.
Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.

a025129 n = a025129_list !! (n-1)
a025129_list= f (tail a000040_list) [head a000040_list] 1 where
   f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014

f[n_] := Block[{primeList = Prime@ Range@ n}, Total[ Take[ primeList, Floor[n/2]]*Reverse@ Take[ primeList, {Floor[(n + 3)/2], n}]]]; Array[f, 44] (* Robert G. Wilson v, Apr 07 2014 *)

A025129=n->sum(k=1,n\2,prime(k)*prime(n-k+1)) \\ M. F. Hasler, Apr 06 2014

a(n) = A024697(n) for even n. - M. F. Hasler, Apr 06 2014

Following suggestions by Robert Israel and N. J. A. Sloane, initial 0=a(1) added by M. F. Hasler, Apr 06 2014

A145518 Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119

Offset: 1

Tilman Neumann, Oct 12 2008

Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e., the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.
Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.
Then the triangle is given by T1[n,k;x] := Sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i is the i-th prime number (A000040).
2nd column (4, 6, 19, 29, 68, 94, 177, ...) is A024697.
Row sums give A145519.

			Triangle starts:
   2;
   3,   4;
   5,   6,   8;
   7,  19,  12,  16;
  11,  29,  38,  24,  32;
  13,  68,  85,  76,  48,  64;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Tilman Neumann, More terms, partition generator and transform implementation.

Cf. A000040, A024697, A145519, A145520, A258323.

g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,
      expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, May 25 2015

g[n_, i_] := g[n, i] = If[n==0 || i==1, (2 x)^n, Expand[Sum[g[n-i*j, i-1]*(Prime[i]*x)^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Reference to more terms etc. changed to make it version independent by Tilman Neumann, Sep 02 2009

A258358 Sum over all partitions lambda of n into 3 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

30, 42, 136, 293, 551, 892, 1765, 2570, 4273, 6747, 9770, 13958, 21206, 28280, 39702, 54913, 72227, 94682, 127095, 160046, 206119, 263581, 327790, 406354, 512372, 616764, 754412, 921169, 1100165, 1314196, 1584835, 1854384, 2191013, 2590565, 3006512, 3495086

Offset: 6

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=3 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 4), polynom)
    end:
a:= n-> coeff(g(n$2), x, 3):
seq(a(n), n=6..60);

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 3];
a /@ Range[6, 60] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

A258359 Sum over all partitions lambda of n into 4 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

210, 330, 852, 1826, 4207, 6595, 13548, 21479, 38905, 59000, 95953, 142843, 231431, 324152, 487361, 683227, 1003028, 1347337, 1907811, 2541970, 3526314, 4597020, 6194948, 7969172, 10618000, 13401580, 17424498, 21875750, 28102737, 34685941, 43856482, 53791587

Offset: 10

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=4 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 5), polynom)
    end:
a:= n-> coeff(g(n$2), x, 4):
seq(a(n), n=10..60);

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 4];
a /@ Range[10, 60] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

A258360 Sum over all partitions lambda of n into 5 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

2310, 2730, 7860, 15606, 35594, 67255, 120061, 201324, 364479, 592991, 1004771, 1530056, 2444073, 3691392, 5610179, 8334486, 12213775, 17529361, 25187765, 35345858, 49999364, 68516285, 94223007, 127478773, 172613052, 230362430, 305639795, 401637665, 527011287

Offset: 15

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 15..1000

Column k=5 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 6), polynom)
    end:
a:= n-> coeff(g(n$2), x, 5):
seq(a(n), n=15..60);

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[g[n - i j, i - 1] (Prime[i] x)^j, {j, 0, Min[1, n/i]}]]];
a[n_] := Coefficient[g[n, n], x, 5];
a /@ Range[15, 60] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

A258361 Sum over all partitions lambda of n into 6 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

30030, 39270, 90300, 177930, 381222, 722434, 1477619, 2309879, 4194446, 6846481, 11667593, 18212397, 30309561, 45149226, 70722044, 105790662, 160115543, 232478684, 346845682, 489561123, 709058342, 994019962, 1405076982, 1932862089, 2705315737, 3653574123

Offset: 21

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 21..1000

Column k=6 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 7), polynom)
    end:
a:= n-> coeff(g(n$2), x, 6):
seq(a(n), n=21..60);

A258362 Sum over all partitions lambda of n into 7 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

510510, 570570, 1436820, 2655870, 5532330, 9757518, 19659886, 34710965, 58356321, 96541978, 161476211, 256683013, 419693431, 647984259, 1021626403, 1536889595, 2332063802, 3443800806, 5133970767, 7443724123, 10827942578, 15520714599, 22052126419, 30994058608

Offset: 28

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..1000

Column k=7 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 8), polynom)
    end:
a:= n-> coeff(g(n$2), x, 7):
seq(a(n), n=28..60);

A258363 Sum over all partitions lambda of n into 8 distinct parts of Product_{i:lambda} prime(i).

Original entry on oeis.org

9699690, 11741730, 27927900, 49533330, 98525490, 170218830, 325872714, 562212782, 1032566057, 1629661685, 2724030632, 4284584225, 6990871609, 10713813287, 17001782121, 25600766613, 39614085330, 58088625761, 87187552970, 126762441906, 186103726454, 266554756593

Offset: 36

Alois P. Heinz, May 27 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..1000

Column k=8 of A258323.

Cf. A000040.

g:= proc(n, i) option remember; convert(series(`if`(n=0, 1,
      `if`(i<1, 0, add(g(n-i*j, i-1)*(ithprime(i)*x)^j
      , j=0..min(1, n/i)))), x, 9), polynom)
    end:
a:= n-> coeff(g(n$2), x, 8):
seq(a(n), n=36..60);

cp's OEIS Frontend

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

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A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

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A025129 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.

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A145518 Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

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A258358 Sum over all partitions lambda of n into 3 distinct parts of Product_{i:lambda} prime(i).

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A258359 Sum over all partitions lambda of n into 4 distinct parts of Product_{i:lambda} prime(i).

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A258360 Sum over all partitions lambda of n into 5 distinct parts of Product_{i:lambda} prime(i).

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A258361 Sum over all partitions lambda of n into 6 distinct parts of Product_{i:lambda} prime(i).

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