A000586 -id:A000586 - cp's OEIS frontend

A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

1, -1, -1, 0, 1, 0, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, -1, 2, 0, -1, -1, 1, -1, 1, -1, 2, 0, 0, -2, 2, -2, 0, 0, 3, -2, 1, -2, 2, -1, 0, -3, 5, -1, 0, -3, 3, -3, 3, -3, 3, -1, 2, -5, 6, -4, 1, -2, 6, -5, 3, -6, 5, -2, 4, -8, 9, -5, 3, -5, 7, -8, 7, -8, 8, -4, 5

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of distinct prime parts (including 1) and the number of partitions of n into an odd number of distinct prime parts (including 1).

Convolution inverse of A034891.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000586, A000607, A034891, A036497, A046675 (partial sums).

nmax = 82; CoefficientList[Series[(1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

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

A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

Original entry on oeis.org

1, 0, 0, 0, 5, 6, 42, 64, 387, 5480, 10461, 113256, 507390, 1071084, 4882635, 44984560, 382362589, 891350154, 7469477771, 33066211100, 78673599501, 649785780710, 2884039365010, 22986956007816, 306912836483025, 1361558306986280, 3519406658042964

Offset: 1

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(5) = 5 because fifth prime number is 11 and we have [3, 2, 2, 2, 2], [2, 3, 2, 2, 2], [2, 2, 3, 2, 2], [2, 2, 2, 3, 2] and [2, 2, 2, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A000040, A000586, A000607, A023360, A056768, A070215, A073610, A098238, A117278, A121303, A219180, A224344, A259254, A265112, A341459.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(ithprime(n), n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2021

Table[SeriesCoefficient[Sum[x^Prime[k], {k, 1, n}]^n, {x, 0, Prime[n]}], {n, 1, 27}]

a(n) = [x^prime(n)] (Sum_{k>=1} x^prime(k))^n.

A316202 Number of integer partitions of n into Fermi-Dirac primes.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 31, 37, 45, 54, 65, 77, 92, 109, 128, 152, 177, 208, 242, 283, 327, 380, 439, 506, 583, 669, 768, 878, 1004, 1144, 1303, 1482, 1681, 1906, 2156, 2438, 2750, 3101, 3490, 3924, 4407, 4942, 5538, 6197, 6929

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(12) = 13 integer partitions of 12 into Fermi-Dirac primes:
(75), (93),
(444), (543), (552), (732),
(3333), (4332), (4422), (5322),
(33222), (42222),
(222222).

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

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,n}],{n,0,nn}]

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

A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

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

Index entries for sequences related to partitions

Cf. A000586, A008578, A036497, A067661, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339383.

s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020

nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] + (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A036497(n) + A298602(n)) / 2.

A339383 Number of partitions of n into an odd number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(21) = 4 because we have [17, 3, 1], [13, 7, 1], [13, 5, 3] and [11, 7, 3].

Index entries for sequences related to partitions

Cf. A000586, A008578, A036497, A067659, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339382.

s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020

nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] - (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A036497(n) - A298602(n)) / 2.

A111902 Number of partitions of n into distinct parts that are primes or squares of primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 4, 4, 4, 6, 4, 8, 5, 9, 7, 10, 9, 11, 12, 12, 15, 14, 17, 17, 20, 20, 23, 24, 26, 28, 30, 32, 35, 36, 40, 41, 46, 47, 52, 54, 58, 62, 65, 71, 73, 80, 82, 90, 93, 101, 104, 113, 117, 125, 132, 139, 148, 154, 165, 171, 183, 191

Offset: 1

Reinhard Zumkeller, Aug 20 2005

nonn

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 3*x^7 + x^8 + 4*x^9 + 2*x^10 + ...
a(12) = #{3^2+3, 7+5, 7+3+2, 5+2^2+3} = 4.

Cf. A000586, A111900, A106244, A111901.

{a(n) = if(n < 0, 0, polcoeff( prod(k=1, primepi(n), (1 + x^prime(k)^2 + x*O(x^n)) * (1 + x^prime(k))), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=1} (1 + x^prime(k))*(1 + x^(prime(k)^2)). - Ilya Gutkovskiy, Dec 26 2016

A184083 Decimal expansion of product_{p=primes} (1+1/2^p).

Original entry on oeis.org

1, 4, 6, 2, 4, 3, 1, 2, 1, 9, 0, 0, 6, 9, 5, 9, 8, 0, 1, 3, 5, 7, 2, 2, 9, 0, 4, 0, 1, 8, 9, 7, 7, 7, 0, 5, 7, 4, 2, 5, 6, 5, 7, 5, 7, 5, 8, 0, 8, 7, 7, 8, 3, 2, 0, 1, 1, 6, 6, 9, 4, 2, 9, 9, 3, 1, 0, 3, 3, 8, 8, 9, 7, 7, 9, 3, 9, 7, 6

Offset: 1

R. J. Mathar, Jan 09 2011

Inverse of the constant A184084.

			(1+1/2^2) *(1+1/2^3) *(1+1/2^5) *(1+1/2^7) *(1+1/2^11)* .... = 1.4624312190069598013...

RealDigits[Times@@(1+1/2^Prime[Range[100]]),10,120][[1]] (* Harvey P. Dale, Sep 28 2013 *)

prodeuler(p=2,1e6,1.+(1.>>p)) \\ Charles R Greathouse IV, Aug 09 2022

Equals product_{p in A000040} (1+2^(-p)) = sum_{n>=0} A000586(n)/2^n.

A281273 Expansion of Product_{j>=1} 1/(1 - x^(Sum_{i=1..j} prime(i))).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 4, 3, 4, 6, 4, 7, 4, 7, 7, 7, 9, 8, 9, 12, 9, 14, 10, 15, 14, 15, 17, 17, 18, 22, 19, 25, 21, 27, 27, 28, 31, 31, 33, 38, 36, 42, 39, 45, 47, 49, 52, 55, 55, 64, 61, 70, 67, 74, 77, 81, 84, 91, 89, 102, 98, 110, 109, 116, 123, 126, 133, 141, 141, 156, 153, 168, 169, 178, 188, 193

Offset: 0

Ilya Gutkovskiy, Jan 18 2017

nonn

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

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

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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, A281274.

nmax = 86; CoefficientList[Series[Product[1/(1 - x^Sum[Prime[i], {i, 1, j}]), {j, 1, nmax}], {x, 0, nmax}], x]

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

A300852 L.g.f.: log(Product_{k>=1} (1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

0, 2, 3, -2, 5, -1, 7, -2, 3, -3, 11, -5, 13, -5, 8, -2, 17, -1, 19, -7, 10, -9, 23, -5, 5, -11, 3, -9, 29, -6, 31, -2, 14, -15, 12, -5, 37, -17, 16, -7, 41, -8, 43, -13, 8, -21, 47, -5, 7, -3, 20, -15, 53, -1, 16, -9, 22, -27, 59, -10, 61, -29, 10, -2, 18, -12, 67, -19, 26, -10, 71, -5, 73, -35, 8

Offset: 1

Ilya Gutkovskiy, Mar 13 2018

			L.g.f.: L(x) = 2*x^2/2 + 3*x^3/3 - 2*x^4/4 + 5*x^5/5 - x^6/6 + 7*x^7/7 - 2*x^8/8 + 3*x^9/9 - 3*x^10/10 + ...
exp(L(x)) = 1 + x^2 + x^3 + 2*x^5 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... + A000586(n)*x^n + ...

Cf. A000040 (fixed points), A000586, A008472, A300893, A300894.

nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}],x] Range[0, nmax]]
nmax = 75; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, PrimeQ[#] &], {n, 75}]

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

a(n) = Sum_{p|n, p prime} p * (-1)^(n/p + 1). [See Mmca prog.]

A302236 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 + x^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

sign

The difference between the number of partitions of n into an even number of nonprime parts and the number of partitions of n into an odd number of nonprime parts.

Convolution of the sequences A000586 and A081362.

Index entries for sequences related to partitions

Cf. A000586, A002095, A018252, A048165, A081362, A096258, A302234.

nmax = 80; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[1/(1 + Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 + x^A018252(k)).

cp's OEIS Frontend

A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

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A299168 Number of ordered ways of writing n-th prime number as a sum of n primes.

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A316202 Number of integer partitions of n into Fermi-Dirac primes.

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A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

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Maple

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A339383 Number of partitions of n into an odd number of distinct primes (counting 1 as a prime).

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Maple

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A111902 Number of partitions of n into distinct parts that are primes or squares of primes.

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A184083 Decimal expansion of product_{p=primes} (1+1/2^p).

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A281273 Expansion of Product_{j>=1} 1/(1 - x^(Sum_{i=1..j} prime(i))).

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