A305830 -id:A305830 - cp's OEIS frontend

A326031 Weight of the set-system with BII-number n.

0, 1, 1, 2, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9

Offset: 0

Gus Wiseman, Jul 20 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets of positive integers has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18. The weight of a set-system is the sum of sizes of its elements (sometimes called its edges).

			The sequence of set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 0..8192

Cf. A000120, A029931, A048793, A061775, A070939, A072639, A116549, A302242, A305830, A326701, A326702, A326703, A326704.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Join@@bpe/@bpe[n]],{n,0,100}]

def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def A326031(n): return sum(i.bit_count() for i in bin_i(n)) # John Tyler Rascoe, Jun 08 2024

a(2^x + ... + 2^z) = w(x + 1) + ... + w(z + 1), where x...z are distinct nonnegative integers and w = A000120. For example, a(6) = a(2^2 + 2^1) = w(3) + w(2) = 3.

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

A305831 Number of connected components of the strict integer partition with FDH number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

			Let f = A050376. The FD-factorization of 1683 is 9*11*17 = f(6)*f(7)*f(10). The connected components of {6,7,10} are {{7},{6,10}}, so a(1683) = 2.

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305832.

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)]]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Length[zsm[FDfactor[n]/.FDrules]],{n,nn}]

A305832 Number of connected components of the n-th FDH set-system.

Original entry on oeis.org

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

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. The n-th FDH set-system is obtained by repeating this factorization on each index s_i.

			Let f = A050376. The FD-factorization of 765 is 5*9*17 or f(4)*f(6)*f(10) = f(4)*f(2*3)*f(2*5) with connected components {{{4}},{{2,3},{2,5}}}, so a(765) = 2.

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305831.

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)]]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>1]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Length[csm[FDfactor[#]/.FDrules&/@(FDfactor[n]/.FDrules)]],{n,nn}]

A326032 a(2^x + ... + 2^z) = w(x) + ... + w(z), where x...z are distinct nonnegative integers and w = A000120.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 3, 3, 4, 4, 4, 4, 5

Offset: 0

Gus Wiseman, Jul 22 2019

nonn

From Robert Israel, Jul 23 2019: (Start)
a(2*n+1)=a(2*n).
a(n)=1 if and only if n > 1 is in A283526. (End)

			For example, a(6) = a(2^2 + 2^1) = w(2) + w(1) = 2.

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A022290 (Fibonacci), A059590 (factorials), A073642, A089625 (primes), A116549, A326031.

Cf. A000120, A029931, A035327, A048793, A070939, A283526, A305830, A326031, A326669, A326702.

Bwt:= proc(n) option remember; convert(convert(n,base,2),`+`) end proc:
f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(L[i]*Bwt(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 23 2019

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[Length/@bpe/@(bpe[n]-1)],{n,0,100}]

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A326031 Weight of the set-system with BII-number n.

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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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A305831 Number of connected components of the strict integer partition with FDH number n.

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A305832 Number of connected components of the n-th FDH set-system.

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A326032 a(2^x + ... + 2^z) = w(x) + ... + w(z), where x...z are distinct nonnegative integers and w = A000120.

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