A000586 -id:A000586 - cp's OEIS frontend

A316210 Number of integer partitions of the n-th Fermi-Dirac prime into Fermi-Dirac primes.

1, 1, 2, 2, 4, 7, 11, 17, 31, 37, 54, 109, 152, 283, 380, 878, 1482, 1906, 3101, 3924, 6197, 11915, 14703, 27063, 40016, 48450, 84633, 101419, 121250, 204461, 398916, 551093, 646073, 883626, 1030952, 1397083, 2522506, 3875455, 5128718, 7741307, 8860676

Offset: 1

Gus Wiseman, Jun 26 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0.

			The a(6) = 7 partitions of 9 into Fermi-Dirac primes are (9), (54), (72), (333), (432), (522), (3222).

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829, A316202, A316211, A316220.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]]
FDprimeList=Select[Range[nn],FDpQ];
ser=Product[1/(1-x^d),{d,FDprimeList}];
Table[SeriesCoefficient[ser,{x,0,FDprimeList[[n]]}],{n,Length[FDprimeList]}]

A316211 Number of strict integer partitions of n into Fermi-Dirac primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 4, 4, 4, 6, 4, 9, 5, 10, 8, 11, 11, 12, 15, 13, 19, 16, 21, 21, 24, 26, 27, 32, 31, 37, 37, 42, 44, 47, 52, 53, 61, 61, 69, 71, 78, 82, 88, 95, 99, 108, 112, 122, 128, 137, 144, 154, 163, 172, 184, 193, 206, 216, 230, 242, 256

Offset: 0

Gus Wiseman, Jun 26 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0.

			The a(16) = 9 strict integer partitions of 16 into Fermi-Dirac primes:
(16),
(9,7), (11,5), (13,3),
(7,5,4), (9,4,3), (9,5,2), (11,3,2),
(7,4,3,2).

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829, A316202, A316210, A316220.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]]
FDprimeList=Select[Range[nn],FDpQ];
ser=Product[1+x^d,{d,FDprimeList}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

O.g.f.: Product_d (1 + x^d) where the product is over all Fermi-Dirac primes (A050376).

A347419 Number of partitions of n into two or more distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 2, 1, 2, 2, 3, 1, 4, 2, 4, 4, 4, 4, 5, 5, 6, 5, 6, 6, 6, 8, 7, 9, 9, 9, 11, 10, 11, 13, 12, 13, 15, 14, 17, 16, 18, 18, 20, 21, 23, 22, 25, 25, 27, 30, 29, 32, 32, 34, 37, 38, 40, 42, 44, 45, 50, 49, 53, 55, 57, 60, 64, 66, 70, 71, 76, 78, 83, 86, 89, 93, 96

Offset: 1

Ayoub Saber Rguez, Aug 31 2021

nonn

Every positive integer can be written as a sum of two or more distinct primes except 1,2,3,4,6 and 11.

			a(5) = 1: 2+3.
a(18) = 4: 11+7, 11+5+2, 13+5, 13+3+2.

Cf. A000040, A000586, A010051, A166081.

h:= proc(n) h(n):=`if`(n<2, 0, `if`(isprime(n), n, h(n-1))) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
      b(n, h(i-1))+b(n-i, h(min(n-i, i-1)))))
    end:
a:= n-> b(n, h(n-1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 03 2021

m = 24; Rest @ CoefficientList[Series[Product[(1 + x^Prime[k]), {k, 1, m}], {x, 0, Prime[m]}], x] - Table[Boole @ PrimeQ[n], {n, 1, Prime[m]}] (* Amiram Eldar, Sep 03 2021 *)

from sympy import isprime, primerange
from functools import cache
@cache
def A000586(n, k=None): # after Charles R Greathouse IV
    if k == None: k = n
    if n < 1: return int(n == 0)
    return sum(A000586(n-p, p-1) for p in primerange(1, k+1))
def a(n): return A000586(n) - isprime(n)
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Sep 03 2021

a(n) = A000586(n) - A010051(n).

A347550 Number of partitions of n into at most 2 distinct prime parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 0, 2, 1, 3, 2, 2, 1, 3, 0, 4, 1, 1, 1, 3, 1, 4, 2, 3, 1, 3, 1, 5, 1, 4, 0, 3, 1, 5, 1, 3, 0, 3, 1, 6, 2, 2, 1, 5, 0, 6, 1, 2, 1, 5, 1, 6, 2, 4, 1, 5, 0, 7, 1, 4, 1, 4, 1, 8, 1, 4

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000586, A117929, A219180, A223893.

a(n) = Sum_{k=0..2} A219180(n,k). - Alois P. Heinz, Sep 08 2021

A358010 Number of partitions of n into at most 5 distinct prime parts.

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, 13, 15, 15, 17, 15, 18, 17, 20, 20, 23, 20, 25, 22, 27, 28, 28, 27, 30, 29, 36, 34, 38, 36, 41, 35, 48, 41, 48, 44, 50, 46, 58, 53, 61, 54, 64, 55, 72, 66, 74

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

nonn

Cf. A000040, A000586, A219180, A219199, A223893, A347550, A347587, A347609, A347742, A358009, A358011.

A358011 Number of partitions of n into at most 6 distinct prime parts.

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, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 31, 35, 36, 39, 40, 42, 42, 45, 49, 50, 52, 55, 53, 61, 61, 67, 67, 70, 70, 77, 77, 86, 84

Offset: 0

Ilya Gutkovskiy, Oct 24 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Robert Israel)

Cf. A000040, A000586, A219180, A219200, A223893, A347550, A347588, A347610, A347743, A358009, A358010.

P:= select(isprime,[2,seq(i,i=3..100,2)]):
G:= mul(1+t*x^p, p=P):
f:= proc(n) local i,S;
   S:= coeff(G,x,n);
   add(coeff(S,t,i),i=0..6)
end proc;
map(f, [$0..100]); # Robert Israel, May 14 2025

a(n) = Sum_{k=0..6} A219180(n,k). - Alois P. Heinz, May 14 2025

A141272 Number of partitions of n into distinct nonzero Mancala numbers (A007952).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 3, 3, 2, 2, 2, 3, 4, 2, 2, 5, 4, 3, 4, 5, 5, 4, 4, 5, 7, 4, 3, 8, 7, 5, 6, 6, 8, 9, 5, 7, 12, 7, 7, 10, 8, 11, 9, 8, 14, 13, 9, 12, 16, 13, 13, 13, 14, 18, 14, 12, 17, 19, 14, 16, 20, 17, 20, 19, 18, 25, 21, 18, 25, 24, 21

Offset: 1

Reinhard Zumkeller, Jun 21 2008

nonn

			a(20) = #{17+3, 11+9, 11+5+3+1} = 3;
a(21) = #{21, 17+3+1, 11+9+1} = 3;
a(22) = #{21+1, 17+5} = 2.

Cf. A141271, A141279, A000586.

A164067 Number of partitions of n into distinct Sophie Germain primes.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 1, 3, 0, 2, 2, 0, 3, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 4, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 4, 2, 3, 3, 2, 2, 3, 1, 2, 3, 1, 3, 3, 2, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 5, 3, 5, 4, 4, 5, 3

Offset: 1

Reinhard Zumkeller, Aug 09 2009

nonn

			e(16) = #{11+5, 11+3+2} = 2.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A164066, A000586, A000009, A005384.

ok(n)={isprime(n) && isprime(2*n+1)}
{my(n=80); Vec(prod(k=1, n, 1 + if(ok(k), x^k + O(x*x^n), 0))-1, -n)} \\ Andrew Howroyd, Dec 28 2017

G.f.: Product_{k>=1} (1 + x^A005384(k)). - Andrew Howroyd, Dec 28 2017

A208614 Number of partitions of n into distinct primes where all of the prime factors of n are represented in the partition.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 4, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 4, 2, 1, 5, 9, 6, 5, 4, 1, 6, 6, 7, 4, 3, 1, 4, 1, 4, 9, 20, 7, 3, 1, 7, 8, 6, 1, 15, 1, 5, 19, 11, 13, 9, 1, 21, 52, 7, 1

Offset: 0

Richard Penner, Feb 29 2012

nonn

Inspired by web-based discussion started by Rajesh Bhowmick.

			a(p) = 1 for any prime p.
a(n) = 0 for 1, 4, 6, 8, 9, 22.
a(25) = 3 because 25 = 3 + 5 + 17 = 5 + 7 + 13 = 2 + 5 + 7 + 11.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Rajesh Bhowmick, A bit different from Goldbach's conjecture (February 27-28, 2012)

Cf. A000586 (upper bound). A000586(A076694(n)) is a stricter upper bound.

with(numtheory):
a:= proc(n) 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([({seq(ithprime(i), i=1..pi(n))} minus f)[]]);
      b(n-add(i, i=f), nops(l))
    end:
seq(a(n), n=0..300);  # Alois P. Heinz, Mar 20 2012

restrictedIntegerPartition[ n_Integer, list_List ] := 1 /; n == 0
restrictedIntegerPartition[ n_Integer, list_List ] := 0 /; n < 0 || Total[list] < n || n < Min[list]
restrictedIntegerPartition[ n_Integer, list_List ] := restrictedIntegerPartition[n - First[list], Rest[list]] + restrictedIntegerPartition[n, Rest[list]]
distinctPrimeFactors[ n_Integer ] := distinctPrimeFactors[n] = Map[First, FactorInteger[n]]
oeisA076694[ n_Integer ] := oeisA076694[n] = n - Total[distinctPrimeFactors[n]]
oeisA208614[ n_Integer ] := restrictedIntegerPartition[oeisA076694[n], Sort[Complement[Prime @ Range @ PrimePi @ oeisA076694 @ n, distinctPrimeFactors[n]] , Greater ]]
Table[oeisA208614[n], {n,1,100}]

countRestrictedIntegerPartitions(n, L) := if ( n = 0 ) then 1 else if ( ( n < 0 ) or ( lsum(k, k, L) < n ) or ( n < lmin( L ) ) ) then 0 else block( [ m, R ], m : first(L), R : rest(L), countRestrictedIntegerPartitions(n, R) + countRestrictedIntegerPartitions(n - m, R));
distinctPrimeFactors(n) := map(first,ifactors(n));
oeisA076694(n) := n - lsum(k,k,distinctPrimeFactors(n));
listOfPrimesLessThanOrEqualTo (n) := block( [ list : [] , i], for i : 2 step 0 while i <= n do ( list : cons(i, list) , i : next_prime(i) ) , list );
oeisA208614(n) := block([ m, list ], m : oeisA076694(n), list : sort(listify(setdifference(setify(listOfPrimesLessThanOrEqualTo(m)), setify(distinctPrimeFactors(n)))), ordergreatp), countRestrictedIntegerPartitions(m, list));
makelist(oeisA208614(j), j, 1, 100);

A281274 Expansion of Product_{j>=1} (1 + x^(Sum_{i=1..j} prime(i))).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 3, 0, 1, 2, 0, 2, 0, 0, 2, 1, 2, 1, 0, 2, 1, 3, 1, 2, 0, 2, 1, 1, 2, 0, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 2, 0, 3, 0, 2, 2, 1, 4, 1, 3, 2, 3, 2, 2, 1, 2, 3

Offset: 0

Ilya Gutkovskiy, Jan 18 2017

nonn

Number of partitions of n into distinct nonzero partial sums of primes (A007504).

			a(17) = 2 because we have [17] and [10, 5, 2], where 2 = prime(1), 5 = prime(1) + prime(2), 10 = prime(1) + prime(2) + prime(3), 17 = prime(1) + prime(2) + prime(3) + prime(4).

Eric Weisstein's World of Mathematics, Prime Sums
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for related partition-counting sequences

Cf. A000586, A000607, A007504, A281273.

nmax = 110; CoefficientList[Series[Product[1 + x^Sum[Prime[i], {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{j>=1} (1 + x^(Sum_{i=1..j} prime(i))).

cp's OEIS Frontend

A316210 Number of integer partitions of the n-th Fermi-Dirac prime into Fermi-Dirac primes.

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A316211 Number of strict integer partitions of n into Fermi-Dirac primes.

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A347419 Number of partitions of n into two or more distinct primes.

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A347550 Number of partitions of n into at most 2 distinct prime parts.

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A358010 Number of partitions of n into at most 5 distinct prime parts.

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A358011 Number of partitions of n into at most 6 distinct prime parts.

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A141272 Number of partitions of n into distinct nonzero Mancala numbers (A007952).

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A164067 Number of partitions of n into distinct Sophie Germain primes.

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PARI

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A208614 Number of partitions of n into distinct primes where all of the prime factors of n are represented in the partition.

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Maxima