A316219 -id:A316219 - cp's OEIS frontend

A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

1, 1, 3, 3, 9, 21, 46, 95, 273, 363, 731, 3088, 6247, 24152, 46012, 319511, 1141923, 2138064, 7346404, 13530107, 45297804, 271446312

Offset: 1

Gus Wiseman, Jun 26 2018

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. An FD-partition is an integer partition of a Fermi-Dirac prime into Fermi-Dirac primes. a(n) is the number of sequences of FD-partitions whose sums are weakly decreasing and sum to the n-th Fermi-Dirac prime.

Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first six terms of A316220.

Cf. A050376, A063834, A064547, A213925, A269134, A281113, A299757, A305829, A316202, A316210, A316211, A316219, A316228.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]];
FDpl=Select[Range[nn],FDpQ];
fen[n_]:=fen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],FDpQ]}],{x,0,n}];
Table[Sum[Times@@fen/@p,{p,Select[IntegerPartitions[FDpl[[n]]],And@@FDpQ/@#&]}],{n,Length[FDpl]}]

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

Original entry on oeis.org

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.

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.

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

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A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

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

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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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A316525 Numbers whose average of prime factors is prime.

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