A000586 -id:A000586 - cp's OEIS frontend

A223893 Number of partitions of n into at most three distinct primes.

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 4, 3, 4, 3, 4, 3, 5, 3, 5, 3, 4, 4, 5, 6, 5, 5, 5, 5, 7, 6, 5, 7, 4, 7, 7, 8, 7, 7, 6, 10, 8, 9, 9, 8, 7, 12, 8, 12, 8, 10, 6, 14, 9, 15, 8, 13, 7, 14, 11, 16, 8, 14, 7, 19, 11, 19, 10, 15, 9, 21, 12, 20, 11, 18

Offset: 1

Frank M Jackson, Mar 28 2013

nonn

The sequence shows a stronger version of the Goldbach conjecture that for n > 6, n has partitions with at most three distinct primes.

			a(21)=3 as 21 = 2+19 = 3+5+13 = 3+7+11.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000586, A000607, A071335, A185101, A218007, A218469.

a[n_] := Length@Select[IntegerPartitions[n, 3, Prime@Range@PrimePi@n],
Sort@#==Union@# &]; Array[a, 100] (* Giovanni Resta, Mar 29 2013 *)

A001823 Central factorial numbers: column 2 in triangle A008956.

Original entry on oeis.org

0, 9, 259, 1974, 8778, 28743, 77077, 179452, 375972, 725781, 1312311, 2249170, 3686670, 5818995, 8892009, 13211704, 19153288, 27170913, 37808043, 51708462, 69627922, 92446431, 121181181, 157000116, 201236140, 255401965, 321205599, 400566474, 495632214

Offset: 1

N. J. A. Sloane

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
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 linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

A bisection of A181888.

Column 2 in triangle A008956.

A001823:=-(9+196*z+350*z**2+84*z**3+z**4)/(z-1)**7; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Table[1/90*n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n + 7), {n, 40}] (* Stefan Steinerberger, Apr 15 2006 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,9,259,1974,8778,28743,77077},30] (* Harvey P. Dale, Jun 09 2013 *)

a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n+7)/90.
If we replace n with n-1/2 in this formula we get 16*A000586(n).
G.f.: z*(9+196*z+350*z**2+84*z**3+z**4)/(1-z)^7.
a(1)=0, a(2)=9, a(3)=259, a(4)=1974, a(5)=8778, a(6)=28743, a(7)=77077, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 09 2013

More terms from Stefan Steinerberger, Apr 15 2006

A112022 Number of partitions of n into distinct Chen primes.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 14, 17, 15, 17, 19, 18, 21, 21, 21, 24, 24, 26, 28, 27, 30, 30, 32, 35, 34, 37, 37, 39, 41, 43, 45, 46, 48, 51, 53, 56, 58, 59, 61, 64, 66, 70, 71, 73

Offset: 0

Reinhard Zumkeller, Aug 26 2005

nonn

a(n) = A000586(n) for n <= 42.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A112021, A112020, A109611.

terms = 81;
gf = Times @@ (1 + x^SequencePosition[ PrimeOmega[ Range[terms]], {1, _, 1|2}][[All, 1]]) + O[x]^terms;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2018 *)

P=1+O(x^1001); forprime(p=2,1e3,if(bigomega(p+2)<3,P*=1+x^p)); Vec(P) \\ Charles R Greathouse IV, May 13 2013

G.f.: Product_{k>=1} (1 + x^A109611(k)). - Andrew Howroyd, Dec 28 2017

A137793 Number of partitions of n into distinct parts with no prime gaps in their factorization.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 31, 38, 43, 50, 58, 66, 76, 88, 100, 113, 130, 146, 165, 188, 211, 237, 267, 298, 334, 375, 417, 464, 517, 573, 636, 706, 781, 862, 954, 1050, 1157, 1276, 1401, 1539, 1689, 1851, 2027, 2222, 2427

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			a(16)=A000009(16)-#{14+2,10+6,10+5+1,10+4+2,10+3+2+1}=32-5=27.

Cf. A073491, A000041, A000586, A106244, A137792, A137791.

A209402 Number of partitions of n into distinct primes except the prime factors of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 4, 3, 2, 3, 1, 2, 6, 3, 8, 4, 2, 5, 3, 5, 10, 4, 3, 5, 13, 4, 14, 7, 4, 7, 18, 9, 12, 9, 6, 9, 25, 11, 10, 10, 9, 14, 34, 10, 38, 16, 11, 24, 14, 13, 49, 20, 18, 12, 60, 25, 66, 31, 17, 28

Offset: 0

Alois P. Heinz, Mar 22 2012

a(n) = 0 for n in {1, 2, 3, 4, 6, 11}, a(n) = 1 for n in {0, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 21, 27}, a(n) = 2 for n in {16, 18, 19, 20, 22, 25, 28, 33}, a(n) = 3 for n in {24, 26, 30, 35, 39}, a(n) = 4 for n in {23, 32, 38, 42, 45}. The smallest n such that a(n) = 0, 1, 2, ... is 1, 0, 16, 24, 23, 34, 29, 44, 31, 48, 37, 54, 49, 41, 43, ... . Missing values are in {15, 19, 21, 22, 26, 33, 35, 36, 37, 39, 42, ... }.

			a(5) = 1: [2,3].
a(7) = 1: [2,5].
a(16) = 2: [3,13], [5,11].
a(23) = 4: [3,7,13], [2,3,5,13], [5,7,11], [2,3,7,11].
a(24) = 3: [5,19], [7,17], [11,13].
a(29) = 6: [3,7,19], [2,3,5,19], [5,7,17], [2,3,7,17], [5,11,13], [2,3,11,13].
a(34) = 5: [3,31], [5,29], [11,23], [3,5,7,19], [3,7,11,13].

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

Cf. A000586, A208614.

with(numtheory):
a:= proc(n) option remember; local b, l, f;
      b:= proc(h, j) option remember;
            `if`(h=0, 1, `if`(j<1, 0,
            `if`(l[j]>h, 0, b(h-l[j], j-1)) +b(h, j-1)))
          end; forget(b);
      f:= factorset(n);
      l:= sort([({ithprime(i)$i=1..pi(n)} minus f)[]]);
      b(n, nops(l))
    end:
seq(a(n), n=0..300);

a[n_] := a[n] = Module[{b, l, f}, b[h_, j_] := b[h, j] = If[h == 0, 1, If[j < 1, 0, If[l[[j]] > h, 0, b[h - l[[j]], j-1]] + b[h, j-1]]]; f = FactorInteger[n][[All, 1]]; l = Sort[Union[Array[Prime, PrimePi[n]] ~Complement~ f]]; b[n, Length[l]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 27 2017, translated from Maple *)

a(p) = A000586(p)-1 for any prime p.

A283876 Number of partitions of n into distinct twin primes (A001097).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 6, 5, 7, 6, 8, 7, 7, 9, 7, 9, 8, 9, 9, 9, 9, 11, 11, 11, 12, 11, 14, 12, 13, 14, 14, 13, 15, 15, 17, 16, 16, 19, 17, 20, 19, 21, 21, 21, 21, 23, 23, 23, 23, 24, 26, 25, 28, 28, 30, 29, 30, 32

Offset: 0

Ilya Gutkovskiy, Mar 17 2017

nonn

			a(29) = 4 because we have [29], [19, 7, 3], [17, 7, 5] and [13, 11, 5].

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A001097, A077608, A129363, A283875.

nmax = 95; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

listA001097(lim)=my(v=List([3]),p=5); forprime(q=7,lim, if(q-p==2, listput(v,p); listput(v,q)); p=q); if(p+2>lim && isprime(p+2), listput(v,p)); Vec(v)
first(n)=my(v=listA001097(n),x=O('x^(n+1))+'x); Vec(prod(i=1,#v, 1+x^v[i]))[1..n+1] \\ Charles R Greathouse IV, Mar 17 2017

Vec(prod(k=1, 95, (1 + (isprime(k) && (isprime(k - 2) || isprime(k + 2)))*x^k)) + O(x^96)) \\ Indranil Ghosh, Mar 17 2017

G.f.: Product_{k>=1} (1 + x^A001097(k)).

A316154 Number of integer partitions of prime(n) into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 12, 19, 39, 50, 93, 136, 166, 239, 409, 682, 814, 1314, 1774, 2081, 3231, 4272, 6475, 11077, 14270, 16265, 20810, 23621, 30031, 68251, 85326, 118917, 132815, 226097, 251301, 342448, 463940, 565844, 759873, 1015302, 1117708, 1787452, 1961624

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

			The a(7) = 9 partitions of 17 into a prime number of prime parts: (13,2,2), (11,3,3), (7,7,3), (7,5,5), (7,3,3,2,2), (5,5,3,2,2), (5,3,3,3,3), (5,2,2,2,2,2,2), (3,3,3,2,2,2,2).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Andrew Howroyd)

Cf. A000040, A000586, A000607, A038499, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316185, A344782.

b:= proc(n, p, c) option remember; `if`(n=0 or p=2,
      `if`(n::even and isprime(c+n/2), 1, 0),
      `if`(p>n, 0, b(n-p, p, c+1))+b(n, prevprime(p), c))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 26 2018

Table[Length[Select[IntegerPartitions[Prime[n]],And[PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,20}]
(* Second program: *)
b[n_, p_, c_] := b[n, p, c] = If[n == 0 || p == 2, If[EvenQ[n] && PrimeQ[c + n/2], 1, 0], If[p>n, 0, b[n - p, p, c + 1]] + b[n, NextPrime[p, -1], c]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 50] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

seq(n)={my(p=vector(n,k,prime(k))); my(v=Vec(1/prod(k=1, n, 1 - x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

a(n) = A085755(A000040(n)). - Alois P. Heinz, Jun 26 2018

Terms a(21) and beyond from Andrew Howroyd, Jun 26 2018

A316185 Number of strict integer partitions of the n-th prime into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 2, 3, 5, 5, 6, 8, 10, 13, 18, 20, 26, 32, 34, 45, 54, 66, 90, 106, 117, 135, 142, 165, 269, 311, 375, 398, 546, 579, 689, 823, 938, 1107, 1301, 1352, 1790, 1850, 2078, 2153, 2878, 3811, 4241, 4338, 4828, 5495, 5637, 7076, 8000, 9032

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

			The a(14) = 8 partitions of 43 into a prime number of distinct prime parts: (41,2), (31,7,5), (29,11,3), (23,17,3), (23,13,7), (19,17,7), (19,13,11), (17,11,7,5,3).

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000586, A000607, A038499, A045450, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316154.

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(isprime(n), n, h(n-1)))
    end:
b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(isprime(c), 1, 0), `if`(i<2, 0, b(n, h(i-1), c)+
      `if`(i>n, 0, b(n-i, h(min(n-i, i-1)), c+1))))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..56);  # Alois P. Heinz, May 26 2021

Table[Length[Select[IntegerPartitions[Prime[n]],And[UnsameQ@@#,PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,10}]
(* Second program: *)
h[n_] := h[n] = If[n == 0, 0, If[PrimeQ[n], n, h[n - 1]]];
b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[PrimeQ[c], 1, 0], If[i < 2, 0, b[n, h[i - 1], c] +
     If[i > n, 0, b[n - i, h[Min[n - i, i - 1]], c + 1]]]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 56] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)

seq(n)={my(p=vector(n, k, prime(k))); my(v=Vec(prod(k=1, n, 1 + x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

a(n) = A045450(A000040(n)).

More terms from Alois P. Heinz, Jun 26 2018

A083290 Number of partitions of n into distinct parts which are coprime to n and which are also pairwise relatively prime.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 7, 2, 9, 3, 4, 5, 16, 3, 20, 4, 8, 7, 31, 5, 22, 9, 18, 9, 54, 4, 68, 16, 21, 16, 28, 11, 112, 20, 32, 18, 144, 9, 173, 22, 33, 40, 221, 19, 139, 25, 71, 43, 327, 25, 117, 47, 103, 80, 475, 18, 568, 90, 98, 122, 191, 29, 805, 93, 197, 44

Offset: 1

Reinhard Zumkeller, Apr 23 2003

nonn

a(n) <= A036998(n); see A082415 for numbers m with a(m) = A036998(m).

			a(7) = 3 since 7 = 3+4 = 2+5 = 1+6; 7 = 1+2+4 does not count (A036998(7)=4).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000010, A000586, A007360.

a[n_] := a[n] = If[n == 1, 1, Module[{ip}, ip = IntegerPartitions[n, All, Select[Range[n - 1], CoprimeQ[#, n] &]]; Length@Select[ip, Sort[#] == Union[#] && AllTrue[Subsets[#, {2}], CoprimeQ @@ # &] &]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 12 2021 *)

a(n)={local(Cache=Map()); my(recurse(r,p,k)=my(hk=[r,p,k],z); if(!mapisdefined(Cache,hk,&z), z=if(k==0, r==0, self()(r,p,k-1) + if(gcd(p,k)==1, self()(r-k, p*k, min(r-k,k-1)))); mapput(Cache, hk, z)); z); recurse(n,n,n)} \\ Andrew Howroyd, Apr 20 2021

Terms a(31) and beyond from Andrew Howroyd, Apr 20 2021

A280245 Expansion of Product_{k>=1} (1 + x^prime(k))^2.

Original entry on oeis.org

1, 0, 2, 2, 1, 6, 1, 8, 6, 6, 14, 6, 18, 14, 18, 24, 23, 30, 35, 38, 46, 54, 55, 74, 72, 90, 100, 106, 128, 136, 152, 178, 185, 216, 238, 252, 302, 308, 359, 390, 420, 478, 512, 564, 628, 668, 745, 810, 871, 974, 1035, 1140, 1238, 1336, 1459, 1586, 1700, 1868, 1993, 2168, 2354, 2512, 2751, 2930, 3177, 3418, 3677, 3960

Offset: 0

Ilya Gutkovskiy, Dec 29 2016

nonn

Number of partitions of n into distinct prime parts, with 2 types of each part.

Self-convolution of A000586. - Ilya Gutkovskiy, Jan 19 2018

			a(5) = 6 because we have [5], [5'], [3, 2], [3', 2], [3, 2'], [3', 2'].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for related partition-counting sequences

Cf. A000009, A000586, A000607, A022567, A070215.

nmax = 67; CoefficientList[Series[Product[(1 + x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x]

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

log(a(n)) ~ 2*Pi*sqrt(n/(3*log(n/2))). - Vaclav Kotesovec, Jan 12 2021

cp's OEIS Frontend

A223893 Number of partitions of n into at most three distinct primes.

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A001823 Central factorial numbers: column 2 in triangle A008956.

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A112022 Number of partitions of n into distinct Chen primes.

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A137793 Number of partitions of n into distinct parts with no prime gaps in their factorization.

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A209402 Number of partitions of n into distinct primes except the prime factors of n.

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Maple

Mathematica

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A283876 Number of partitions of n into distinct twin primes (A001097).

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PARI

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A316154 Number of integer partitions of prime(n) into a prime number of prime parts.

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Maple

Mathematica

PARI

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A316185 Number of strict integer partitions of the n-th prime into a prime number of prime parts.

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