A368109 -id:A368109 - cp's OEIS frontend

A371450 MM-number of the set-system with BII-number n.

1, 3, 5, 15, 13, 39, 65, 195, 11, 33, 55, 165, 143, 429, 715, 2145, 29, 87, 145, 435, 377, 1131, 1885, 5655, 319, 957, 1595, 4785, 4147, 12441, 20735, 62205, 47, 141, 235, 705, 611, 1833, 3055, 9165, 517, 1551, 2585, 7755, 6721, 20163, 33605, 100815, 1363, 4089

Offset: 0

Gus Wiseman, Apr 02 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The set-system with BII-number 30 is {{2},{1,2},{3},{1,3}} with MM-number prime(3) * prime(6) * prime(5) * prime(10) = 20735.
The terms together with their prime indices and binary indices of prime indices begin:
     1 -> {}        -> {}
     3 -> {2}       -> {{1}}
     5 -> {3}       -> {{2}}
    15 -> {2,3}     -> {{1},{2}}
    13 -> {6}       -> {{1,2}}
    39 -> {2,6}     -> {{1},{1,2}}
    65 -> {3,6}     -> {{2},{1,2}}
   195 -> {2,3,6}   -> {{1},{2},{1,2}}
    11 -> {5}       -> {{3}}
    33 -> {2,5}     -> {{1},{3}}
    55 -> {3,5}     -> {{2},{3}}
   165 -> {2,3,5}   -> {{1},{2},{3}}
   143 -> {5,6}     -> {{1,2},{3}}
   429 -> {2,5,6}   -> {{1},{1,2},{3}}
   715 -> {3,5,6}   -> {{2},{1,2},{3}}
  2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}

The sorted version is A329629, with empties A302494.
A019565 gives Heinz number of binary indices.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326753 counts connected components for BII-numbers, ones A326749.
Cf. A000720, A003963, A087086, A096111, A275024, A302242, A302505, A302521, A326782, A329557, A329630, A368109.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]),{n,0,30}]

A371454 Numbers whose binary indices are all semiprimes.

Original entry on oeis.org

8, 32, 40, 256, 264, 288, 296, 512, 520, 544, 552, 768, 776, 800, 808, 8192, 8200, 8224, 8232, 8448, 8456, 8480, 8488, 8704, 8712, 8736, 8744, 8960, 8968, 8992, 9000, 16384, 16392, 16416, 16424, 16640, 16648, 16672, 16680, 16896, 16904, 16928, 16936, 17152

Offset: 1

Gus Wiseman, Apr 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   768:     1100000000 ~ {9,10}
   776:     1100001000 ~ {4,9,10}
   800:     1100100000 ~ {6,9,10}
   808:     1100101000 ~ {4,6,9,10}

Partitions of this type are counted by A101048, squarefree case A002100.
For primes instead of semiprimes we get A326782.
For prime indices instead of binary indices we have A339112, A339113.
The squarefree case is A371453.
A001358 lists semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A296119, A302478, A326031, A367905, A368109, A368533, A371450.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
semi[n_]:=PrimeOmega[n]==2;
Select[Range[10000],And@@semi/@bix[#]&]

from math import isqrt
from sympy import primepi, primerange
def A371454(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A001358(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

Original entry on oeis.org

0, 1, 4, 5, 256, 257, 260, 261, 67108864, 67108865, 67108868, 67108869, 67109120, 67109121, 67109124, 67109125, 1208925819614629174706176, 1208925819614629174706177, 1208925819614629174706180, 1208925819614629174706181, 1208925819614629174706432

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For powers of 2 instead of 3 we have A253317.

			The terms together with their binary expansions and binary indices begin:
         0:                           0 ~ {}
         1:                           1 ~ {1}
         4:                         100 ~ {3}
         5:                         101 ~ {1,3}
       256:                   100000000 ~ {9}
       257:                   100000001 ~ {1,9}
       260:                   100000100 ~ {3,9}
       261:                   100000101 ~ {1,3,9}
  67108864: 100000000000000000000000000 ~ {27}
  67108865: 100000000000000000000000001 ~ {1,27}
  67108868: 100000000000000000000000100 ~ {3,27}
  67108869: 100000000000000000000000101 ~ {1,3,27}
  67109120: 100000000000000000100000000 ~ {9,27}
  67109121: 100000000000000000100000001 ~ {1,9,27}
  67109124: 100000000000000000100000100 ~ {3,9,27}
  67109125: 100000000000000000100000101 ~ {1,3,9,27}

Michael De Vlieger, Table of n, a(n) for n = 1..256

A000244 lists powers of 3.
A048793 lists binary indices, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A058891, A062050, A072639, A253317, A326031, A326675, A326702, A367912, A368183, A368109.

Select[Range[0,10000],IntegerQ[Log[3,Times@@Join@@Position[Reverse[IntegerDigits[#,2]],1]]]&]
(* Second program *)
{0}~Join~Array[FromDigits[Reverse@ ReplacePart[ConstantArray[0, Max[#]], Map[# -> 1 &, #]], 2] &[3^(Position[Reverse@ IntegerDigits[#, 2], 1][[;; , 1]] - 1)] &, 255] (* Michael De Vlieger, Dec 29 2023 *)

a(3^n) = 2^(3^n - 1).

A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 45, 1352, 157647, 63380093, 85147722812, 385321270991130

Offset: 0

Gus Wiseman, Mar 12 2024

			The set-system {{2},{1,2},{2,4},{1,3,4}} has unique choice (2,1,4,3) so is counted under a(4).

This is the unique version of A367902, complement A367903.
Choosing a sequence gives A367904, ranks A367908.
The maximal case is A368601, complement A368600.
This is the restriction of A370638 to A000225.
Factorizations of this type are counted by A370645.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A133686, A134964, A367772, A367867, A368101, A368109, A370584.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Union[Sort/@Select[Tuples[#],UnsameQ@@#&]]]==1&]],{n,0,3}]

a(n) = A370638(2^n - 1).

Binomial transform of A368601. - Christian Sievers, Aug 12 2024

a(5)-a(8) from Christian Sievers, Aug 12 2024

A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 10, 15, 35, 56, 84, 120, 165, 220, 286, 364, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398, 850668

Offset: 0

Gus Wiseman, Mar 11 2024

nonn

			The a(1) = 1 through a(7) = 15 subsets:
  {}  {1}  {1}  {1,2}  {1,2}  {1,2}  {1,2}
           {2}  {1,3}  {1,3}  {1,3}  {1,3}
                {2,3}  {1,4}  {1,4}  {1,4}
                       {2,3}  {1,5}  {1,5}
                       {2,4}  {2,3}  {1,6}
                       {3,4}  {2,4}  {2,3}
                              {2,5}  {2,4}
                              {3,4}  {2,5}
                              {3,5}  {2,6}
                              {4,5}  {3,4}
                                     {3,5}
                                     {3,6}
                                     {4,5}
                                     {4,6}
                                     {5,6}

The version without subtracting one is A357812.
Dominates A370641, see also A370640.
A007318 counts subsets by cardinality.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A029837, A113473, A326031, A367905, A368109, A370636.

Table[If[n==0,0,Binomial[n-1,IntegerLength[n,2]-1]],{n,0,15}]

a(n) = binomial(n - 1, A029837(n+1) - 1) = binomial(n - 1, A113473(n) - 1) = binomial(n - 1, A070939(n) - 1) for n > 0.

cp's OEIS Frontend

A371450 MM-number of the set-system with BII-number n.

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A371454 Numbers whose binary indices are all semiprimes.

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A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

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A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

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A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.

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