A279065 -id:A279065 - cp's OEIS frontend

A299757 Weight of the strict integer partition with FDH number n.

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

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)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]

A061773 Triangle in which n-th row lists Matula-Goebel numbers for all rooted trees with n nodes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 15, 18, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 53, 59, 67, 25, 27, 30, 33, 35, 36, 39, 40, 42, 44, 46, 47, 48, 49, 51, 52, 56, 57, 58, 61, 62, 64, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 101, 106

Offset: 1

N. J. A. Sloane, Jun 22 2001

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.
n-th row has A000081(n) terms.
First entry in row n is A005517(n).
Last entry in row n is A005518(n).
The Maple program yields row n after defining F = A005517(n) and L = A005518(n).

			The labels for the rooted trees with at most 4 nodes are as follows (x is the root):
                                         o
                                         |
               o         o        o   o  o
               |          \        \ /   |
     o  o   o  o  o o o    o   o    o    o
     |   \ /   |   \|/      \ /     |    |
  x  x    x    x    x        x      x    x
  1  2    4    3    8        6      7    5 (label)
Triangle begins:
1;
2;
3,4;
5,6,7,8;
9,10,11,12,13,14,16,17,19;
15,18,20,21,22,23,24,26,28,29,31,32,34,37,38,41,43,53,59,67;
25,27,30,33,35,36,39,40,42,44,46,47,48,49,51,52,56,57,58,61,62,64,68,\
71,73,74,76,79,82,83,86,89,101,106,107,109,118,127,131,134,139,157,163,\
179,191,241,277,331;
...
Triangle of rooted trees represented as finitary multisets begins:
(),
(()),
((())), (()()),
(((()))), (()(())), ((()())), (()()()),
((())(())), (()((()))), ((((())))), (()()(())), ((()(()))), (()(()())), (()()()()), (((()()))), ((()()())). - _Gus Wiseman_, Dec 21 2016

Alois P. Heinz, Rows n = 1..12, flattened
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP code calculating rows
Gus Wiseman, Matula-Goebel trees triangle illustration n=1..6
Index entries for sequences related to Matula-Goebel numbers
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A061775 (number of nodes), A000081 (row lengths), A005517 (row minimum), A005518 (row maximum), A214572 (row n=8).
Cf. A347620 (inverse permutation).
Cf. A276625, A279065, A279614.

n := 8: F := 45: L := 2221: with(numtheory): N := proc (m) local r, s: r := proc (m) options operator, arrow: op(1, factorset(m)) end proc: s := proc (m) options operator, arrow: m/r(m) end proc: if m = 1 then 1 elif bigomega(m) = 1 then 1+N(pi(m)) else N(r(m))+N(s(m))-1 end if end proc: A := {}: for k from F to L do if N(k) = n then A := `union`(A, {k}) else  end if end do: A;

F[n_] := F[n] = Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^(n/3-1), Mod[n, 3] == 1, 5^(n/3-1/3), True, 9*5^(n/3-5/3)]; L[n_] := L[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, , Prime[L[n-1]]]; r[m] := FactorInteger[m][[1, 1]]; s[m_] := m/r[m]; NN[m_] := NN[m] = Which[m == 1, 1, PrimeOmega[m] == 1, 1+NN[PrimePi[m]], True, NN[r[m]]+NN[s[m]]-1]; row[n_] := Module[{A, k}, A = {}; For[k = F[n], k <= L[n], k++, If[NN[k] == n, A = Union[A, {k}]]]; A]; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Maple *)
nn=8;MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
Take[GatherBy[Range[Switch[nn,1,1,2,2,3,4,,Nest[Prime,8,nn-4]]],MGweight],nn] (* _Gus Wiseman, Dec 21 2016 *)

PARI
```
\\ See links.
```

More terms from Emeric Deutsch, May 01 2004

A279861 Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 2, 2, 5, 4, 6, 8, 10, 14, 23, 26, 34, 46, 64, 81, 115, 158, 199, 277, 376, 505, 684, 934, 1241, 1711, 2300, 3123, 4236, 5763, 7814, 10647, 14456, 19662

Offset: 1

Gus Wiseman, Dec 21 2016

nonn

A finitary set is transitive if every element is also a subset. Transitive sets are also called full sets.

			Sequence of transitive finitary sets begins:
1  ()
2  (())
4  (()(()))
7  (()(())((())))
8  (()(())(()(())))
11 (()(())((()))(((()))))
   (()(())((()))(()(())))
12 (()(())((()))(()((()))))
13 (()(())((()))((())((()))))
   (()(())(()(()))((()(()))))
14 (()(())((()))(()(())((()))))
   (()(())(()(()))(()(()(()))))
15 (()(())((()))(((())))(()(())))
   (()(())(()(()))((())(()(()))))
16 (()(())((()))(((())))((((())))))
   (()(())((()))(((())))(()((()))))
   (()(())((()))(()(()))(()((()))))
   (()(())((()))(()(()))((()(()))))
   (()(())(()(()))(()(())(()(()))))
17 (()(())((()))(((())))(()(((())))))
   (()(())((()))(((())))((())((()))))
   (()(())((()))(()(()))(()(()(()))))
   (()(())((()))(()(()))((())((()))))
18 (()(())((()))(((())))((())(((())))))
   (()(())((()))(((())))(()(())((()))))
   (()(())((()))(()(()))((())(()(()))))
   (()(())((()))(()(()))(()(())((()))))
   (()(())((()))((()((()))))(()((()))))
   (()(())((()))(()((())))((())((()))))

Wikipedia, Transitive set
Gus Wiseman, Transitive rooted identity trees example n=23

Cf. A001192, A004111, A061773, A279614, A276625, A279065, A279863.

transfins[n_]:=transfins[n]=If[n===1,{{}},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[nov,_List,{0,Infinity}]<=n]]]/@#,#]&,Array[transfins,n-1,1,Union]],Count[#,_List,{0,Infinity}]===n&]];
Table[Length[transfins[n]],{n,20}]

A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 6, 8, 5, 8, 9, 10, 6, 11, 12, 10, 7, 12, 6, 13, 8, 12, 15, 14, 8, 15, 9, 14, 10, 20, 18, 16, 11, 16, 12, 17, 10, 18, 21, 24, 12, 19, 18, 20, 13, 20, 24, 21, 12, 28, 15, 22, 14, 22, 24, 23, 15, 30, 27, 32, 14, 24, 30, 24, 20, 25

Offset: 1

Gus Wiseman, Jun 10 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = s_1 * ... * s_k.

Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Tree of x -> a(x) for n = 1...75

Cf. A003963, A050376, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A305830, A305831, A305832.

nn=100;
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)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Times@@(FDfactor[n]/.FDrules),{n,nn}]

\\ here isfd is membership test for A050376.
isfd(n)={my(e=isprimepower(n)); e && e == 1<Andrew Howroyd, Aug 02 2018

A279614 a(1)=1, a(d(x_1)..d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 5, 5, 6, 7, 6, 6, 7, 7, 6, 7, 6, 8, 8, 8, 8, 7, 7, 7, 7, 7, 9, 8, 8, 8, 7, 9, 8, 10, 8, 7, 9, 8, 9, 8, 9, 7, 10, 9, 8, 9, 8, 9, 8, 9, 9, 9, 8, 11, 10, 10, 9, 9, 10, 8, 9, 10, 9, 10, 10, 8, 10, 9, 11, 8, 9, 8, 8, 9, 11, 12, 9, 8, 10, 10, 9

Offset: 1

Gus Wiseman, Dec 15 2016

nonn

A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.

In analogy with the Matula-Goebel correspondence between rooted trees and positive integers (see A061775), the iterated normalized Fermi-Dirac representation gives a correspondence between rooted identity trees and positive integers. Then a(n) is the number of nodes in the rooted identity tree corresponding to n.

			Sequence of rooted identity trees represented as finitary sets begins:
{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
{{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
{{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
{{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
{{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.

OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065.

nn=200;
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)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];
FDweight[n_?(#<=nn&)]:=If[n===1,1,1+Total[FDweight[Position[FDprimeList,#][[1,1]]]&/@FDfactor[n]]];
Array[FDweight,nn]

Number of appearances of n is |a^{-1}(n)| = A004111(n).

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 12, 9, 14, 15, 24, 11, 18, 20, 21, 30, 13, 22, 27, 28, 40, 42, 16, 26, 33, 35, 36, 54, 56, 60, 17, 32, 39, 44, 45, 66, 70, 72, 84, 120, 19, 34, 48, 52, 55, 63, 78, 88, 90, 105, 108, 168, 23, 38, 51, 64, 65, 77, 96, 104, 110, 126

Offset: 1

Gus Wiseman, Feb 18 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

This sequence is a permutation of the positive integers.

			Triangle of strict partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (51) (42) (321)
        (7) (61) (43) (52) (421)
     (8) (71) (62) (53) (431) (521)
(9) (81) (72) (54) (63) (621) (531) (432).

Cf. A050376, A056239, A064547, A106400, A122111, A213925, A215366, A246867, A279065, A279614, A299755, A299757, A299758.

Row lengths give A000009.

nn=25;
FDprimeList=Select[Range[nn],MatchQ[FactorInteger[#],{{?PrimeQ,?(MatchQ[FactorInteger[2#],{{2,_}}]&)}}]&];
Table[Sort[Times@@FDprimeList[[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,Length[FDprimeList]}]

A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 1, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 3, 3, 2, 3

Offset: 1

Gus Wiseman, Jun 10 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = w(s_1) + ... + w(s_k) where w = A064547.

			Sequence of FDH set-systems (a list containing all finite sets of finite sets of positive integers) begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{2}}
   5: {{3}}
   6: {{},{1}}
   7: {{4}}
   8: {{},{2}}
   9: {{1,2}}
  10: {{},{3}}
  11: {{5}}
  12: {{1},{2}}
  13: {{1,3}}
  14: {{},{4}}
  15: {{1},{3}}
  16: {{6}}
  17: {{1,4}}
  18: {{},{1,2}}
  19: {{7}}
  20: {{2},{3}}
  21: {{1},{4}}
  22: {{},{5}}
  23: {{2,3}}
  24: {{},{1},{2}}
  25: {{8}}
  26: {{},{1,3}}
  27: {{1},{1,2}}

Cf. A003963, A050376, A056239, A061775, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A302242, A302243, A305829, A305832.

nn=100;
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)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[Length/@(FDfactor/@(FDfactor[n]/.FDrules))],{n,nn}]

A316202 Number of integer partitions of n into Fermi-Dirac primes.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 31, 37, 45, 54, 65, 77, 92, 109, 128, 152, 177, 208, 242, 283, 327, 380, 439, 506, 583, 669, 768, 878, 1004, 1144, 1303, 1482, 1681, 1906, 2156, 2438, 2750, 3101, 3490, 3924, 4407, 4942, 5538, 6197, 6929

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(12) = 13 integer partitions of 12 into Fermi-Dirac primes:
(75), (93),
(444), (543), (552), (732),
(3333), (4332), (4422), (5322),
(33222), (42222),
(222222).

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829.

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,n}],{n,0,nn}]

O.g.f.: Product_d 1/(1 - x^d) where the product is over all Fermi-Dirac primes (A050376).

A322027 Maximum order of primeness among the prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 1, 2, 3, 4, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 4, 1, 2, 3, 1, 2, 1, 1, 3, 5, 1, 4, 2, 3, 2, 1, 1, 2, 3, 2, 2, 1, 4, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 4, 1, 2, 1, 3, 3, 1, 5, 2, 1, 3, 4, 2, 2, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factor of 105 = 3*5*7 with maximum order of primeness is 5, with order 3.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322028, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> max(0, map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Max@@(Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]])],{n,100}]

A279863 Number of maximal transitive finitary sets with n brackets.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 2, 1, 1, 4, 3, 4, 2, 5, 6, 10, 8, 11, 11, 20, 22, 29, 36, 45, 53, 77, 83, 108, 141, 172, 208, 274, 323

Offset: 1

Gus Wiseman, Dec 21 2016

nonn

A finitary set is transitive if every element is also a subset. A set system is maximal if the union is also a member.

			The a(23)=3 maximal transitive finitary sets are:
(()(())(()(()))((())(()(())))(()(())(()(())))),
(()(())((()))(((())))(()((())))(()(())((())))),
(()(())((()))(()(()))(()((())))(()(())((())))).

Gus Wiseman, Maximal transitive identity trees up to n=25

Cf. A001192, A004111, A276625, A279065, A279614, A279861.

maxtransfins[n_]:=If[n===1,{},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[Union[nov,{Union@@nov}],_List,{0,Infinity}]<=n]]]/@#,#]&,{{}}],And[Count[#,_List,{0,Infinity}]===n,MemberQ[#,Union@@#]]&]];
Table[Length[maxtransfins[n]],{n,20}]

cp's OEIS Frontend

A299757 Weight of the strict integer partition with FDH number n.

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Mathematica

A061773 Triangle in which n-th row lists Matula-Goebel numbers for all rooted trees with n nodes.

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Maple

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PARI

Extensions

A279861 Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.

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Mathematica

A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.

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Mathematica

PARI

A279614 a(1)=1, a(d(x_1)*..*d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.

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Mathematica

Formula

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

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Mathematica

A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).

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Mathematica

A316202 Number of integer partitions of n into Fermi-Dirac primes.

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A279614 a(1)=1, a(d(x_1)..d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.