A316220 -id:A316220 - cp's OEIS frontend

A316223 Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.

0, 1, 1, 4, 1, 6, 1, 13, 4, 6, 1, 25, 1, 6, 6, 38, 1, 26, 1, 26, 6, 6

Offset: 1

Gus Wiseman, Jun 27 2018

A positive subset-sum is a pair (h,g), where h is a positive integer and g is an integer partition, such that some submultiset of g sums to h. A triangle consists of a root sum r and a sequence of positive subset-sums ((h_1,g_1),...,(h_k,g_k)) such that the sequence (h_1,...,h_k) is weakly decreasing and has a submultiset summing to r. The composite of a triangle is (r, g_1 + ... + g_k) where + is multiset union.

			We write positive subset-sum triangles in the form rootsum(branch,...,branch). The a(8) = 13 triangles:
  1(1(1,1,1))
  2(2(1,1,1))
  3(3(1,1,1))
  1(1(1),1(1,1))
  2(1(1),1(1,1))
  1(1(1),2(1,1))
  2(1(1),2(1,1))
  3(1(1),2(1,1))
  1(1(1,1),1(1))
  2(1(1,1),1(1))
  1(1(1),1(1),1(1))
  2(1(1),1(1),1(1))
  3(1(1),1(1),1(1))

Cf. A063834, A262671, A269134, A276024, A281113, A299701, A301934, A301935, A316219, A316220, A316222.

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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).

A316222 Number of positive subset-sum triangles whose composite is a positive subset-sum of an integer partition of n.

Original entry on oeis.org

1, 5, 20, 74, 258, 855, 2736, 8447

Offset: 1

Gus Wiseman, Jun 27 2018

A positive subset-sum is a pair (h,g), where h is a positive integer and g is an integer partition, such that some submultiset of g sums to h. A triangle consists of a root sum r and a sequence of positive subset-sums ((h_1,g_1),...,(h_k,g_k)) such that the sequence (h_1,...,h_k) is weakly decreasing and has a submultiset summing to r.

			We write positive subset-sum triangles in the form rootsum(branch,...,branch). The a(2) = 5 positive subset-sum triangles:
  2(2(2))
  1(1(1,1))
  2(2(1,1))
  1(1(1),1(1))
  2(1(1),1(1))

Cf. A063834, A262671, A269134, A276024, A281113, A301934, A301935, A316219, A316220.

A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 28, 29, 31, 34, 36, 37, 39, 40, 41, 43, 46, 47, 48, 49, 52, 53, 55, 56, 58, 59, 61, 63, 66, 67, 71, 73, 76, 79, 81, 82, 83, 88, 89, 90, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112

Offset: 1

Gus Wiseman, Jun 27 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. Every positive integer has a unique factorization into distinct Fermi-Dirac primes.

			Sequence of multiarrows in the form "number: sum <= factors" begins:
   2:  2 <= {2}
   3:  3 <= {3}
   4:  4 <= {4}
   5:  5 <= {5}
   6:  5 <= {2,3}
   7:  7 <= {7}
   9:  9 <= {9}
  10:  7 <= {2,5}
  11: 11 <= {11}
  12:  7 <= {3,4}
  13: 13 <= {13}
  14:  9 <= {2,7}
  16: 16 <= {16}
  17: 17 <= {17}
  18: 11 <= {2,9}
  19: 19 <= {19}
  20:  9 <= {4,5}
  22: 13 <= {2,11}
  23: 23 <= {23}
  24:  9 <= {2,3,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050376, A064547, A100118, A213925, A299757, A305829, A316202, A316210, A316211, A316220.

FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
Select[Range[2,200],Length[FDfactor[Total[FDfactor[#]]]]==1&]

cp's OEIS Frontend

A316223 Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.

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

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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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A316222 Number of positive subset-sum triangles whose composite is a positive subset-sum of an integer partition of n.

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A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

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Mathematica