A056768 -id:A056768 - cp's OEIS frontend

A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

1, 1, 3, 6, 15, 31, 92, 161, 464, 2347, 3987, 18202, 50136, 81722, 214976, 903048, 3684567, 5842249, 23206424, 57341256, 89938662, 343306266, 829972421, 3084219358, 17375700038, 40920517008, 62656899579, 146415515992, 223442878751, 518427758704, 9544240589455, 21746920337606

Offset: 1

Gus Wiseman, Jun 26 2018

nonn

A prime partition is an integer partition of a prime number into prime parts. Then a(n) is the number of sequences of prime partitions whose sums are weakly decreasing and sum to the n-th prime number.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first five terms of A316219.

Cf. A000040, A000041, A000607, A056768, A063834, A100118, A269134, A281113, A316220.

nn=20;
pen[n_]:=pen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],PrimeQ]}],{x,0,n}]
Table[Sum[Times@@pen/@p,{p,Select[IntegerPartitions[Prime[n]],And@@PrimeQ/@#&]}],{n,nn}]

P(n,f)={1/prod(k=1, n, 1 - f(k)*x^prime(k) + O(x*x^prime(n)))}
seq(n)={my(p=P(n, i->1), q=P(n, i->polcoef(p, prime(i)))); vector(n, k, polcoef(q, prime(k)))} \\ Andrew Howroyd, Jan 16 2023

Terms a(16) and beyond from Andrew Howroyd, Jan 16 2023

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.

A316091 Heinz numbers of integer partitions of prime numbers.

Original entry on oeis.org

3, 4, 5, 6, 8, 11, 14, 15, 17, 18, 20, 24, 26, 31, 32, 33, 35, 41, 42, 44, 45, 50, 54, 56, 58, 59, 60, 67, 69, 72, 74, 80, 83, 92, 93, 95, 96, 106, 109, 114, 119, 122, 124, 127, 128, 141, 143, 145, 152, 153, 157, 158, 161, 170, 174, 177, 179, 182, 188, 191

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

Also the union of prime-indexed rows of A215366.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Sequence of all integer partitions of prime numbers begins (2), (1, 1), (3), (2, 1), (1, 1, 1), (5), (4, 1), (3, 2), (7), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (6, 1).

Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316092.

primeMS[n_] := If[n == 1, {}, Flatten[Cases[FactorInteger[n],{p_, k_} :> Table[PrimePi[p], {k}]]]]; Select[Range[100], PrimeQ[Total[primeMS[#]]] &]

A168470 Number of partitions of prime(n) into prime parts smaller than itself.

Original entry on oeis.org

0, 0, 1, 2, 5, 8, 16, 22, 39, 86, 110, 218, 335, 412, 613, 1082, 1849, 2197, 3629, 5006, 5860, 9281, 12487, 19231, 33438, 43708, 49870, 64670, 73505, 94624, 221264, 279515, 394169, 441249, 766261, 853691, 1175343, 1608013, 1975107, 2675924, 3605665, 3977860

Offset: 1

Juri-Stepan Gerasimov, Nov 26 2009

nonn

The difference between this sequence and A056768 is that the number itself (unpartitioned) is not counted.

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

Cf. A000040, A056768.

a(n) = A056768(n) - 1. - R. J. Mathar, Nov 27 2009

a(7) corrected and sequence extended by R. J. Mathar, Nov 27 2009

A276687 Number of prime plane trees of weight prime(n).

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 122, 336, 1412, 15129, 44561, 417542, 2479120, 7540843, 35983502, 451454834, 5313515136, 16809858904, 190077477328, 1124302066470, 3521811953565, 38563707677633, 240966297786218, 3192420711942298, 95433674596402663, 567734580765228356

Offset: 1

Gus Wiseman, Sep 13 2016

nonn

A prime plane tree is either (case 1) a prime number, or (case 2) a sequence of prime plane trees whose weights are an integer partition of a prime number, where the weight of a tree is the sum of weights of its branches. Prime plane trees are "multichains" in the multiorder of integer partitions of prime numbers into prime parts (A056768).

			The a(5) = 11 prime plane trees of weight A000040(5) = 11 are: {11, (3,3,5), (3,3,(2,3)), (2,2,7), (2,2,(2,5)), (2,2,(2,(2,3))), (2,2,(2,2,3)), (2,3,3,3), (2,2,2,5), (2,2,2,(2,3)), (2,2,2,2,3)}.

Alois P. Heinz, Table of n, a(n) for n = 1..681
Gus Wiseman, Comcategories and Multiorders, (pdf version)

Cf. A000040, A056768, A000607, A100118, A196545, A273873.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=2, 0,
       b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)*(1+
      `if`(i>2, b(i, prevprime(i)), 0))))
    end:
a:= n-> `if`(n<3, 1, 1+b(ithprime(n), ithprime(n-1))):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 15 2016

n=20;
ser=Product[1/(1-c[Prime[i]]*x^Prime[i]),{i,1,n}];
sys=Table[c[Prime[i]]==Expand[SeriesCoefficient[ser,{x,0,Prime[i]}]-c[Prime[i]]+1],{i,1,n}];
Block[{c},Set@@@sys]

A316153 Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.

Original entry on oeis.org

15, 33, 45, 93, 153, 177, 275, 327, 369, 405, 425, 537, 603, 605, 775, 831, 891, 1025, 1059, 1125, 1413, 1445, 1475, 1641, 1705, 1719, 1761, 2057, 2075, 2319, 2511, 2577, 2979, 3175, 3179, 3189, 3459, 3485, 3603, 3609, 3825, 3925, 4299, 4475, 4497, 4565, 4581

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of integer partitions of prime numbers into a prime number of prime parts, preceded by their Heinz numbers, begins:
   15: (3,2)
   33: (5,2)
   45: (3,2,2)
   93: (11,2)
  153: (7,2,2)
  177: (17,2)
  275: (5,3,3)
  327: (29,2)
  369: (13,2,2)
  405: (3,2,2,2,2)
  425: (7,3,3)
  537: (41,2)
  603: (19,2,2)
  605: (5,5,3)
  775: (11,3,3)
  831: (59,2)
  891: (5,2,2,2,2)

Cf. A000586, A000607, A038499, A056239, A056768, A064688, A070215, A085755, A302590, A316092, A316151.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[PrimeQ[PrimeOmega[#]],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A316151 Heinz numbers of strict integer partitions of prime numbers into prime parts.

Original entry on oeis.org

3, 5, 11, 15, 17, 31, 33, 41, 59, 67, 83, 93, 109, 127, 157, 177, 179, 191, 211, 241, 277, 283, 327, 331, 353, 367, 401, 431, 461, 509, 537, 547, 563, 587, 599, 617, 709, 739, 773, 797, 831, 859, 877, 919, 967, 991, 1031, 1059, 1063, 1087, 1153, 1171, 1201

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of strict integer partitions of prime numbers into prime parts, preceded by their Heinz numbers, begins:
   3: (2)
   5: (3)
  11: (5)
  15: (3,2)
  17: (7)
  31: (11)
  33: (5,2)
  41: (13)
  59: (17)
  67: (19)
  83: (23)
  93: (11,2)

Cf. A000586, A000607, A038499, A056239, A056768, A064688, A070215, A085755, A302590, A316092.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SquareFreeQ[#],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A316525 Numbers whose average of prime factors is prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 44, 47, 49, 53, 57, 59, 60, 61, 64, 67, 68, 69, 71, 73, 79, 81, 83, 85, 89, 93, 97, 101, 103, 105, 107, 109, 112, 113, 116, 121, 125, 127, 128, 129, 131, 133, 137, 139

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

Prime factors counted with multiplicity. - Harvey P. Dale, Sep 28 2018

			60 = 2*2*3*5 has average of prime factors (2+2+3+5)/4 = 3, which is prime, so 60 belongs to the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A078175.

Cf. A000040, A000607, A056768, A071321, A100118, A316219, A316313.

Select[Range[100],PrimeQ[Mean[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]]&]
Select[Range[200],PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Sep 28 2018 *)

isok(n) = {my(f=factor(n)); iferr(isprime(sum(k=1, #f~, f[k,1]*f[k,2])/sum(k=1, #f~, f[k,2])), E, 0);} \\ Michel Marcus, Jul 06 2018

A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 25, 9, 61, 91, 99, 151, 901, 303, 1759, 3379, 5239, 4713, 8227, 12901, 12537, 23059, 65239, 159421, 232369, 489817, 351237, 726295, 564363, 1101883, 2517865, 6916027, 11825821, 4942227, 27166753, 21280053, 39547957, 52630273, 113638975

Offset: 1

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(4) = 3 because we have [7], [5, 2] and [2, 5].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to compositions

Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(ithprime(n), n, 0):
seq(a(n), n=1..42);  # Alois P. Heinz, Jan 31 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, Function[p, If[p > n, 0, b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];
a[n_] := b[Prime[n], n, 0];
Array[a, 42] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

a(n) = A219107(A000040(n)).

A096066 Triangle read by rows, 1<=k<=n: T(n,k) is the number of occurrences of the k-th prime in partitions of the n-th prime into primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 10, 6, 2, 1, 1, 16, 9, 4, 2, 1, 1, 37, 22, 11, 6, 2, 1, 1, 54, 32, 15, 9, 3, 2, 1, 1, 107, 65, 32, 19, 7, 5, 2, 1, 1, 266, 165, 84, 50, 22, 15, 7, 5, 2, 1, 353, 219, 112, 69, 30, 21, 10, 7, 3, 1, 1, 779, 487, 254, 157, 73, 52, 27, 19, 10, 3, 2, 1, 1270, 795, 420, 261, 124, 90, 49, 36, 19, 7, 5, 1, 1

Offset: 1

Reinhard Zumkeller, Jul 21 2004

			n=5, A000040(5)=11 with A056768(5)=6 partitions into primes:
T(5,1)=10 prime(1)=2 in 7+2+2=5+2+2+2=3+3+3+2=3+2+2+2+2,
T(5,2)=6 prime(2)=3: in 5+3+3=3+3+3+2=3+2+2+2+2,
T(5,3)=2 prime(3)=5: in 5+3+3=5+2+2+2,
T(5,4)=1 prime(4)=7: in 7+2+2.
Triangle begins:
  1;
  0,  1;
  1,  1, 1;
  3,  1, 1, 1;
  10, 6, 2, 1, 1;
  ...

Cf. A056768.

ip[p_] := ip[p] = IntegerPartitions[p, All, Select[Range[p], PrimeQ]] // Flatten;
T[n_, k_] := Count[ip[Prime[n]], Prime[k]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 23 2021 *)

T(n,n) = 1.

Name modified by Jean-François Alcover, Sep 23 2021

cp's OEIS Frontend

A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

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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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A316091 Heinz numbers of integer partitions of prime numbers.

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A168470 Number of partitions of prime(n) into prime parts smaller than itself.

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A276687 Number of prime plane trees of weight prime(n).

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A316153 Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.

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A316151 Heinz numbers of strict integer partitions of prime numbers into prime parts.

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A316525 Numbers whose average of prime factors is prime.

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