A343753
Number of partitions of prime(n) containing a prime number of primes.
Original entry on oeis.org
0, 0, 2, 6, 26, 49, 154, 258, 666, 2404, 3588, 11096, 22477, 31620, 61247, 157725, 387527, 518155, 1208470, 2086019, 2726745, 5975695, 9935799, 20882243, 54355088, 86547260, 108874661, 171286370, 214236058, 333331046, 1486031972, 2246585402, 4132451733
Offset: 1
a(4) = 6 because there are 6 partitions of prime(4) = 7 that contain a prime number of primes (including repetitions). These partitions are [5,2], [3,3,1], [3,2,2], [3,2,1,1], [2,2,2,1], [2,2,1,1,1].
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nterms=20;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,IntegerPartitions[Prime[n]]]],{n,1,nterms}]
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forprime(p=2, 67, my(m=0); forpart(X=p, my(j=0); for(k=1, #X, if(isprime(X[k]), j++));if(isprime(j),m++)); print1(m, ", ")) \\ Hugo Pfoertner, May 01 2021
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
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].
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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 *)
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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
A236019
Smallest number having at least n partitions that contain at least n primes.
Original entry on oeis.org
0, 2, 5, 8, 10, 13, 15, 17, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130
Offset: 0
a(4) = 10: [2,2,2,2,1,1], [2,2,2,2,2], [3,2,2,2,1], [3,3,2,2].
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b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
`if`(i<1, 0, b(n, i-1, t) +`if`(i>n, 0,
b(n-i, i, t -`if`(t>0 and isprime(i), 1, 0)))))
end:
a:= proc(n) option remember; local k;
for k from a(n-1) while b(k, k, n)Alois P. Heinz, Jan 18 2014
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$RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t - If[t>0 && PrimeQ[i], 1, 0]]]]]; a[n_] := a[n] = Module[{k}, For[k = a[n-1], b[k, k, n] < n, k++]; k]; a[0] = 0; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)
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
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].
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nterms=20;Table[Total[Map[If[Count[#, _?PrimeQ]>0,1,0] &,IntegerPartitions[Prime[n]]]],{n,1,nterms}]
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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
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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
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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
A355225
Number of partitions of n that contain more prime parts than nonprime parts.
Original entry on oeis.org
0, 0, 1, 1, 1, 3, 3, 5, 7, 9, 14, 19, 23, 34, 46, 56, 77, 99, 126, 164, 208, 260, 336, 416, 520, 654, 809, 995, 1237, 1514, 1856, 2274, 2761, 3354, 4078, 4918, 5931, 7153, 8572, 10272, 12298, 14663, 17469, 20787, 24643, 29210, 34568, 40797, 48113, 56664, 66573
Offset: 0
For n = 8 the partitions of 8 that contain more prime parts than nonprime parts are [5, 3], [3, 3, 2], [4, 2, 2], [2, 2, 2, 2], [5, 2, 1], [3, 2, 2, 1], [2, 2, 2, 1, 1]. There are seven of these partitions so a(8) = 7.
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a(n) = my(nb=0); forpart(p=n, if (#select(isprime, Vec(p)) > #p/2, nb++)); nb; \\ Michel Marcus, Jun 25 2022
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from sympy import isprime
from sympy.utilities.iterables import partitions
def c(p): return 2*sum(p[i] for i in p if isprime(i)) > sum(p.values())
def a(n): return sum(1 for p in partitions(n) if c(p))
print([a(n) for n in range(51)]) # Michael S. Branicky, Jun 28 2022
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
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).
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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
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nterms=50;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[n],{1}]]],{n,0,nterms-1}]
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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
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).
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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
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nterms=22;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[Prime[n]],{1}]]],{n,1,nterms}]
A355193
Number of partitions of n that contain at least one odd prime as a part.
Original entry on oeis.org
0, 0, 0, 1, 1, 3, 4, 8, 10, 17, 22, 35, 45, 67, 86, 123, 156, 216, 273, 369, 463, 613, 765, 997, 1236, 1587, 1958, 2485, 3049, 3830, 4677, 5823, 7077, 8740, 10576, 12971, 15629, 19046, 22862, 27701, 33125, 39928, 47579, 57078, 67788, 80963, 95852, 114023
Offset: 0
For n = 6 the partitions of 6 that contain at least one odd prime as a part are [3, 3], [5, 1], [3, 2, 1], [3, 1, 1, 1]. There are four of these partitions so a(6) = 4.
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a(n) = my(nb=0); forpart(p=n, if (#select(x->((x>2) && isprime(x)), Vec(p)) >=1, nb++);); nb; \\ Michel Marcus, Jun 23 2022
A348588
Number of partitions of n into 3 parts with at least 1 prime part.
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 2, 4, 4, 6, 7, 8, 10, 12, 13, 17, 17, 19, 22, 25, 26, 31, 31, 36, 37, 42, 43, 49, 49, 56, 56, 64, 63, 72, 70, 80, 79, 87, 87, 99, 94, 107, 105, 116, 114, 126, 123, 139, 134, 148, 145, 158, 155, 173, 166, 184, 178, 195, 189, 211, 202, 222, 215, 236, 226, 250, 242
Offset: 0
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b:= proc(n, i, t) option remember; series(
`if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),
`if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 4)
end:
a:= n-> coeff(b(n$2, 0), x, 3):
seq(a(n), n=0..66); # Alois P. Heinz, Oct 24 2021
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Table[Length@Select[IntegerPartitions[k,{3}],Or@@PrimeQ@#&],{k,0,66}] (* Giorgos Kalogeropoulos, Oct 24 2021 *)
Showing 1-9 of 9 results.