A058698 -id:A058698 - cp's OEIS frontend

A058697 P(p(n)), P = primes (A000040), p = partition numbers (A000041).

2, 2, 3, 5, 11, 17, 31, 47, 79, 113, 181, 263, 389, 547, 761, 1049, 1453, 1951, 2659, 3511, 4643, 6073, 7933, 10243, 13249, 16981, 21713, 27551, 34841, 43853, 55147, 68863, 85819, 106397, 131779, 162473, 199889, 245039, 300233, 365513

Offset: 0

N. J. A. Sloane, Dec 31 2000

nonn

Prime number whose index is the n-th partition number. - Omar E. Pol, Aug 05 2011

Harvey P. Dale, Table of n, a(n) for n = 0..100

Cf. A058698.

Prime[PartitionsP[Range[0,40]]] (* Harvey P. Dale, Sep 21 2014 *)

a(n)={prime(numbpart(n))} \\ Andrew Howroyd, Dec 28 2017

a(n) = A000040(A000041(n)). - Omar E. Pol, Aug 05 2011

A344677 Number of partitions of n containing a prime number of primes and an arbitrary number of nonprimes.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 6, 9, 13, 20, 26, 36, 49, 68, 90, 120, 154, 201, 258, 330, 418, 532, 666, 834, 1041, 1290, 1592, 1958, 2404, 2935, 3588, 4345, 5278, 6366, 7692, 9215, 11096, 13230, 15853, 18831, 22477, 26580, 31620, 37247, 44145, 51851, 61247, 71681, 84445

Offset: 0

Paolo Xausa, May 26 2021

nonn

			a(6) = 4 because there are 4 partitions of 6 that contain a prime number of primes (including repetitions). These partitions are [3,3], [3,2,1], [2,2,2], [2,2,1,1].

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000040, A058698, A085755, A235945, A343753.

nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,IntegerPartitions[n]]],{n,0,nterms-1}]
(* Second program: *)
seq[n_] := Module[{p}, p = 1/Product[1 - If[PrimeQ[k], y*x^k, 0] + O[x]^n, {k, 2, n}]; CoefficientList[Sum[If[PrimeQ[k], Coefficient[p, y, k], 0], {k, 2, n}]/QPochhammer[x + O[x]^n]/(p /. y -> 1), x]];
seq[50] (* Jean-François Alcover, May 27 2021, after Andrew Howroyd *)

seq(n)={my(p=1/prod(k=2, n, 1 - if(isprime(k), y*x^k) + O(x*x^n))); Vec(sum(k=2, n, if(isprime(k), polcoef(p,k,y)))/eta(x+O(x*x^n))/subst(p,y,1), -(n+1))} \\ Andrew Howroyd, May 26 2021

A343813 Number of partitions of prime(n) containing at least one prime.

Original entry on oeis.org

1, 2, 5, 12, 48, 88, 269, 450, 1176, 4355, 6558, 20958, 43412, 61733, 122194, 324532, 820827, 1107647, 2652517, 4655220, 6133664, 13751210, 23192039, 49730098, 132657130, 213646624, 270244858, 429702432, 540212859, 848899870, 3905568236, 5952945182, 11078643138

Offset: 1

Paolo Xausa, Apr 30 2021

nonn

			a(4) = 12 because there are 12 partitions of prime(4) = 7 that contain at least one prime. These partitions are [7], [5,2], [5,1,1], [4,3], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1], [3,1,1,1,1], [2,2,2,1], [2,2,1,1,1], [2,1,1,1,1,1].

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A058698, A235945.

nterms=20;Table[Total[Map[If[Count[#, _?PrimeQ]>0,1,0] &,IntegerPartitions[Prime[n]]]],{n,1,nterms}]

forprime(p=2,59,my(m=0); forpart(X=p, for(k=1,#X, if(isprime(X[k]),m++;break))); print1(m,", ")) \\ Hugo Pfoertner, Apr 30 2021

seq(n)={my(p=primes(n), m=p[#p]); vecextract(Vec(1/eta(x+O(x*x^m)) - 1/prod(k=1, m, 1-if(!isprime(k), x^k) + O(x*x^m)), -m), p)} \\ Andrew Howroyd, Apr 30 2021

from sympy.utilities.iterables import partitions
from sympy import sieve, prime
def A343813(n):
    p = prime(n)
    pset = set(sieve.primerange(2,p+1))
    return sum(1 for d in partitions(p) if len(set(d)&pset) > 0) # Chai Wah Wu, May 01 2021

a(n) = A235945(A000040(n)).

A182739 First differences of the partition numbers of the primes.

Original entry on oeis.org

2, 1, 4, 8, 41, 45, 196, 193, 765, 3310, 2277, 14795, 22946, 18678, 61493, 205177, 501889, 289685, 1558184, 2017516, 1488484, 7662961, 9489819, 26657456, 83235005, 81250196, 56767824, 159900439, 110796851, 309430388, 3062487667, 2050675209, 5133105512

Offset: 1

Omar E. Pol, Jan 27 2011

nonn

This is the first prime 2, followed by the first differences of the partition numbers of primes.

			a(5) = A000041(A000040(5)) - A000041(A000040(4)) = A000041(11) - A000041(7) = 56 - 15 = 41.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A058698.

with(combinat):
a:= n-> `if`(n=1, 2, (x->x[1]-x[2])(map(numbpart@ithprime, [n, n-1]))):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 27 2011

Range[40] // Prime // PartitionsP // Differences // Prepend[#, 2]& (* Jean-François Alcover, Feb 21 2017 *)

a(1) = 2; a(n) = A000041(A000040(n)) - A000041(A000040(n-1)) for n>1.

More terms from Alois P. Heinz, Jan 27 2011

A193830 Even partition numbers of prime numbers.

Original entry on oeis.org

2, 56, 490, 6842, 124754, 831820, 13848650, 133230930, 214481126, 271248950, 541946240, 851376628, 5964539504, 11097645016, 37027355200, 45060624582, 142798995930, 207890420102, 625846753120, 1820701100652, 3068829878530, 37561133582570, 114540884553038

Offset: 1

Omar E. Pol, Aug 06 2011

nonn

			The even number 56 is in the sequence as the partition number of the prime number 11.

Even numbers in A058698.

Cf. A000040, A000041, A052001, A154796, A154798, A163997, A193831.

Select[PartitionsP[Prime[Range[200]]],EvenQ] (* Harvey P. Dale, Jun 20 2015 *)

A193831 Odd partition numbers of prime numbers.

Original entry on oeis.org

3, 7, 15, 101, 297, 1255, 4565, 21637, 44583, 63261, 329931, 1121505, 2679689, 4697205, 6185689, 23338469, 49995925, 431149389, 3913864295, 13610949895, 80630964769, 362326859895, 749474411781, 2168627105469, 3646072432125, 10085065885767, 27152408925615

Offset: 1

Omar E. Pol, Aug 06 2011

nonn

			The odd number 101 is in the sequence as the partition number of the prime number 13.

Odd numbers in A058698.

Cf. A000040, A000041, A052003, A154795, A163998, A193830.

A247087 a(n) = pi(phi(p(P(n)))) = A000720(A000010(A000041(A000040(n)))).

Original entry on oeis.org

0, 1, 3, 4, 9, 25, 41, 39, 168, 462, 442, 1939, 2571, 3998, 5123, 17040, 24853, 38887, 195022, 183430, 404386, 381060, 1162366, 2105509, 1799881, 5966593, 5380661, 14184985, 10473967, 22631261, 135452589, 109540327, 244730051, 487610708, 604467085, 671043205, 3350187738

Offset: 1

Alois P. Heinz, Mar 14 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..90 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000010, A000040, A000041, A000720, A058697, A058698, A070804.

with(numtheory): with(combinat): p:=numbpart: P:=ithprime:
a:= n-> pi(phi(p(P(n)))):
seq(a(n), n=1..20);

a[n_] := PrimePi @ EulerPhi @ PartitionsP @ Prime @ n;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 25 2017 *)

a(n) = A070804(A058698(n)) = A000720(A000010(A000041(A000040(n)))).

a(31)-a(37) from Amiram Eldar, Sep 03 2024

A342621 Sum of the partition number of the prime factors of n with multiplicity.

Original entry on oeis.org

0, 2, 3, 4, 7, 5, 15, 6, 6, 9, 56, 7, 101, 17, 10, 8, 297, 8, 490, 11, 18, 58, 1255, 9, 14, 103, 9, 19, 4565, 12, 6842, 10, 59, 299, 22, 10, 21637, 492, 104, 13, 44583, 20, 63261, 60, 13, 1257, 124754, 11, 30, 16, 300, 105, 329931, 11, 63, 21, 493, 4567, 831820

Offset: 1

Eric Desbiaux, Mar 16 2021

nonn

			For n = 408 = 2^3*3*17, a(408) = 3 * A000041(2) + A000041(3) + A000041(17) = 3*2 + 3 + 297 = 306.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 3000 terms from Eric Desbiaux)

Cf. A000041, A027746, A058698, A001222.

a:= n-> add(combinat[numbpart](i[1])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 17 2021

{0}~Join~Array[Total@ Map[#2 PartitionsP[#1] & @@ # &, FactorInteger[#]] &, 58, 2] (* Michael De Vlieger, Mar 17 2021 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]*numbpart(f[k,1])); \\ Michel Marcus, Mar 17 2021

def a(n):
    return sum([Partitions(primefactor).cardinality() for (primefactor,exponent) in factor(n) for _ in range(exponent)])
[a(n) for n in (1..100)]

a(A003586(n)) - A001414(A003586(n)) = 0.
a(A006899(n)) * A008480(A006899(n)) - A001414(A006899(n)) = 0.
a(n) = Sum_{k=1..A001222(n)} A000041(A027746(n,k)). - Alois P. Heinz, Apr 09 2021

A344715 Number of partitions of n containing a prime number of distinct primes and an arbitrary number of nonprimes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 12, 20, 27, 42, 56, 80, 107, 151, 195, 265, 342, 453, 577, 753, 949, 1220, 1525, 1930, 2398, 3006, 3701, 4594, 5625, 6922, 8426, 10291, 12455, 15117, 18203, 21955, 26326, 31576, 37689, 45002, 53498, 63581, 75313, 89125, 105199, 124056

Offset: 0

Paolo Xausa, May 27 2021

nonn

			a(10) = 12 because there are 12 partitions of 10 that contain a prime number of primes (not counting repetitions). These partitions are [7,3] (containing 2 primes), [7,2,1] (containing 2 primes), [5,3,2] (containing 3 primes), [5,3,1,1] (containing 2 primes), [5,2,2,1] (containing 2 distinct primes), [5,2,1,1,1] (containing 2 primes), [4,3,2,1] (containing 2 primes), [3,3,2,2] (containing 2 distinct primes), [3,3,2,1,1] (containing 2 distinct primes), [3,2,2,2,1] (containing 2 distinct primes), [3,2,2,1,1,1] (containing 2 distinct primes) and [3,2,1,1,1,1,1] (containing 2 primes).

Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344890.

b:= proc(n, i) option remember; expand(
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1)
          *add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0),
             i=2..degree(p)))(b(n$2)):
seq(a(n), n=0..49);  # Alois P. Heinz, Nov 14 2021

nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[n],{1}]]],{n,0,nterms-1}]

seq(n)={my(p=prod(k=2, n, 1 - y + y/(1 - if(isprime(k), x^k))  + O(x*x^n) ) ); Vec(sum(k=2, n, if(isprime(k), polcoef(p,k,y)))/eta(x+O(x*x^n))/subst(p, y, 1), -(n+1))} \\ Andrew Howroyd, May 27 2021

A344890 Number of partitions of prime(n) containing a prime number of distinct primes and an arbitrary number of nonprimes.

Original entry on oeis.org

0, 0, 1, 3, 20, 42, 151, 265, 753, 3006, 4594, 15117, 31576, 45002, 89125, 235501, 589613, 792426, 1871442, 3251293, 4261819, 9403682, 15690192, 33111688, 86520382, 137957345, 173655404, 273492399, 342231447, 532915031, 2380864800, 3601147053, 6628703864

Offset: 1

Paolo Xausa, Jun 01 2021

nonn

			a(4) = 3 because there are 3 partitions of prime(4)=7 that contain a prime number of primes (not counting repetitions). These partitions are [5,2] (containing 2 primes), [3,2,2] (containing 2 unique primes) and [3,2,1,1] (containing 2 primes).

Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344715.

b:= proc(n, i) option remember; expand(
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1)
          *add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0),
             i=2..degree(p)))(b(ithprime(n)$2)):
seq(a(n), n=1..33);  # Alois P. Heinz, Nov 14 2021

nterms=22;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[Prime[n]],{1}]]],{n,1,nterms}]

a(n) = A344715(A000040(n)).

a(23)-a(33) from Alois P. Heinz, Jun 02 2021

cp's OEIS Frontend

A058697 P(p(n)), P = primes (A000040), p = partition numbers (A000041).

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A344677 Number of partitions of n containing a prime number of primes and an arbitrary number of nonprimes.

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A343813 Number of partitions of prime(n) containing at least one prime.

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A182739 First differences of the partition numbers of the primes.

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A193830 Even partition numbers of prime numbers.

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A193831 Odd partition numbers of prime numbers.

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A247087 a(n) = pi(phi(p(P(n)))) = A000720(A000010(A000041(A000040(n)))).

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A342621 Sum of the partition number of the prime factors of n with multiplicity.

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