A024939 -id:A024939 - cp's OEIS frontend

A099773 Number of partitions of n into odd prime parts.

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 9, 10, 11, 12, 14, 15, 17, 20, 21, 24, 26, 29, 33, 35, 40, 44, 47, 53, 58, 64, 70, 77, 84, 91, 101, 110, 120, 130, 142, 155, 168, 184, 199, 215, 234, 254, 275, 298, 323, 348, 376, 407, 439, 474, 511, 551, 592

Offset: 0

Vladeta Jovovic, Nov 11 2004

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000607, A024939.

a099773 = p a065091_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, Aug 05 2012

CoefficientList[ Series[ Product[1/(1 - x^Prime[i]), {i, 2, 25}], {x, 0, 70}], x] (* Robert G. Wilson v, Jun 14 2006 *)

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

A002124 Number of compositions of n into a sum of odd primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721

Offset: 0

N. J. A. Sloane

nonn

Arises in studying the Goldbach conjecture.

The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

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).

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [The sequence i_n]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Goldbach conjecture

Cf. A002125, A023360, A024939, A077608.

Cf. A065091.

import Data.List (genericIndex)
a002124 n = genericIndex a002124_list n
a002124_list = 1 : f 1 [] a065091_list where
   f x qs ps'@(p:ps)
     | p <= x    = f x (p:qs) ps
     | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
-- Reinhard Zumkeller, Mar 21 2014

A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)]; # N. J. A. Sloane, after MacMahon, Dec 03 2006; used in A002125

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)

a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]

G.f.: 1/( 1 - Sum_{k>=2} x^A000040(k) ). [Joerg Arndt, Sep 30 2012]

Better description and more terms from Philippe Flajolet, Nov 11 2002

Edited by N. J. A. Sloane, Dec 03 2006

A331981 Number of compositions (ordered partitions) of n into distinct odd primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 13, 6, 12, 28, 18, 28, 19, 6, 25, 52, 24, 54, 30, 56, 31, 98, 156, 102, 37, 104, 157, 150, 276, 150, 175, 154, 288, 200, 528, 246, 307, 226, 666, 990, 780, 1038, 679, 348, 799, 1828, 1272, 1162, 1164

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(16) = 4 because we have [13, 3], [11, 5], [5, 11] and [3, 13].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A002124, A023360, A024939, A065091, A099773, A219107, A331926, A331982.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n+1)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i+1))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..72);  # Alois P. Heinz, Feb 03 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n + 1] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, If[# > n, 0, b[n - #, i - 1, t + 1]]&[Prime[i + 1]] + b[n, i - 1, t]]];
a[n_] := b[n, PrimePi[n], 0];
a /@ Range[0, 72] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

A024937 a(n) = number of 2's in all partitions of n into distinct primes.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3, 5, 4, 4, 5, 5, 6, 6, 5, 7, 7, 7, 8, 8, 9, 8, 9, 11, 11, 10, 12, 12, 13, 14, 14, 16, 15, 16, 17, 19, 20, 20, 20, 22, 24, 23, 26, 27, 27, 28, 30, 33, 34, 34, 36, 37, 40, 41, 43, 46, 46, 47, 50, 55, 56, 56, 58, 63, 64

Offset: 0

Clark Kimberling

nonn

Sean A. Irvine, Java program (github)

Cf. A024939.

with(numtheory):
b:= proc(n, i) option remember; local g;
    if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i-1));
         b(n, i-1) +g +[0, `if`(i=1,g[1],0)]
    fi
end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=0..80);  # Sean A. Irvine, after Alois P. Heinz, Jul 29 2019

max = 100; (* number of terms *)
CoefficientList[x^2*Product[1+x^Prime[k], {k, 2, PrimePi[max]}]+O[x]^max, x] (* Jean-François Alcover, Sep 07 2022, after Vladeta Jovovic *)

G.f.: x^2*Product_{k>1} (1+x^prime(k)). - Vladeta Jovovic, Jul 20 2003

More terms from Vladeta Jovovic, Jul 20 2003

a(0)-a(6) prepended by Sean A. Irvine, Jul 29 2019

A298604 Number of partitions of n into distinct odd prime parts (including 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 5, 5, 6, 6, 7, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 14, 14, 15, 16, 17, 17, 17, 20, 22, 21, 22, 24, 25, 27, 28, 30, 31, 31, 33, 36, 39, 40, 40, 42, 46, 47, 49, 53, 54, 55, 58, 63, 67, 68, 70, 73, 77, 81, 84

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Index entries for related partition-counting sequences

Cf. A000586, A006005, A008578, A024939, A036497, A298603.

nmax = 78; CoefficientList[Series[(1 + x) Product[(1 + x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

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

A317678 Sets of n distinct primes p_i in ascending order, written as triangle, such that S_n = Sum_{k=1..n} p_k is minimized and that there exists a compensation set of n primes q_j with q_j = np_j - Sum_{k=1..n,k!=j} p_k, j=1..n such that the set union {p_j} U {q_j} contains 2n distinct primes and Sum_{k=1..n} q_k = S_n.

Original entry on oeis.org

1, 11, 17, 37, 43, 61, 29, 31, 37, 41, 67, 71, 73, 83, 101, 97, 101, 103, 107, 113, 139, 179, 181, 191, 199, 211, 223, 241, 223, 227, 229, 233, 239, 257, 269, 283, 223, 227, 229, 233, 239, 241, 251, 257, 317, 347, 349, 353, 359, 367, 373, 397, 401, 421, 443

Offset: 1

Hugo Pfoertner, Aug 10 2018

In case of ties, i.e., if more than one set exists for the same minimal sum S_n, the lexicographically least set is chosen. Since the condition of distinctness of {p_j} U {q_j} cannot be satisfied for n=1, the sets p = q = {1} are assumed to complete the triangle.

The minimal sums S_n are provided in A317680 and the corresponding compensation sets are provided in A317679. The compensation set is a solution to the "fair compensation" problem. n persons who own individual shares of p_i units of an asset agree to equally share those assets among themselves and a person n+1, who owns 0 units of this asset, but is willing to pay S_n units of compensation, e.g. money, to buy his share of S_n/(n+1) units of the asset. S_n/(n+1) doesn't need to be integer. q_j as defined above is a fair compensation for person j's relinquishment of assets by equipartitioning, assuming a constant price per asset.

			Table begins:
  n  Sum          p_k                       q_k
     A317680                                A317679
  1     1     1                        1
  2    28    11  17                    5  23
  3   141    37  43  61                7  31 103
  4   138    29  31  37  41            7  17  47  67
  5   395    67  71  73  83 101        7  31  43 103 211
  6   660    97 101 103 107 113 139   19  47  61  89 131 313

IBM Research, Ponder This August 2018 - Challenge, 9th row of table is one of many solutions.

Cf. A024939, A317679, A317680.

A369708 Maximal coefficient of (1 + x^3) * (1 + x^5) * (1 + x^7) * ... * (1 + x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 4, 5, 8, 14, 23, 40, 70, 126, 221, 394, 711, 1290, 2354, 4344, 8015, 14868, 27585, 51094, 95160, 178436, 335645, 634568, 1202236, 2261052, 4267640, 8067296, 15318171, 29031484, 55248527, 105251904, 200711160, 383580180, 733704990

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A024939, A025591, A350457.

b:= proc(n) option remember; `if`(n<2, 1, expand(b(n-1)*(1+x^ithprime(n)))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 29 2024

Table[Max[CoefficientList[Product[(1 + x^Prime[k]), {k, 2, n}], x]], {n, 0, 40}]

a(n) = vecmax(Vec(prod(i=2, n, 1+x^prime(i)))); \\ Michel Marcus, Jan 29 2024

from collections import Counter
from sympy import prime
def A369708(n):
    c = {0:1}
    for i in range(2,n+1):
        p, d = prime(i), Counter(c)
        for k in c:
            d[k+p] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 3, 4, 1, 4, 4, 4, 6, 4, 7, 6, 6, 8, 9, 8, 10, 6, 13, 12, 12, 14, 15, 16, 16, 18, 23, 22, 19, 24, 24, 30, 28, 30, 33, 34, 34, 40, 44, 46, 44, 46, 58, 56, 60, 64, 65, 68, 70, 80, 86, 88, 87, 94, 101, 112, 114, 116, 125, 130, 132, 148, 159, 162, 163, 168, 190, 196

Offset: 1

Ilya Gutkovskiy, Jan 23 2017

nonn

Total number of parts in all partitions of n into distinct odd primes.

			a(23) = 7 because we have [23], [13, 7, 3], [11, 7, 5] and 1 + 3 + 3 = 7.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for related partition-counting sequences

Cf. A024938, A024939, A065091.

nmax = 80; Rest[CoefficientList[Series[Sum[x^Prime[k]/(1 + x^Prime[k]), {k, 2, nmax}] Product[1 + x^Prime[k], {k, 2, nmax}], {x, 0, nmax}], x]]

sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + O(x*x^n))*prod(k=1, n, 1+pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ Andrew Howroyd, Dec 28 2017

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

A352166 Number of partitions of n into distinct odd prime powers (1 included).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 12, 12, 13, 14, 16, 17, 17, 19, 21, 23, 23, 25, 28, 30, 31, 33, 37, 38, 40, 43, 47, 50, 52, 55, 60, 64, 66, 70, 76, 81, 83, 89, 96, 101, 105, 110, 119, 125, 130, 138, 147, 155, 161

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A024939, A054685, A061345, A106244, A134337, A280152, A352165.

nmax = 70; CoefficientList[Series[Product[(1 + Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^A061345(k)).

cp's OEIS Frontend

A099773 Number of partitions of n into odd prime parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A002124 Number of compositions of n into a sum of odd primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A331981 Number of compositions (ordered partitions) of n into distinct odd primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A024937 a(n) = number of 2's in all partitions of n into distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A298604 Number of partitions of n into distinct odd prime parts (including 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A369708 Maximal coefficient of (1 + x^3) * (1 + x^5) * (1 + x^7) * ... * (1 + x^prime(n)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).

Views

Author

Keywords

Comments

Examples

Links