cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Apr 02 2024

Keywords

Comments

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.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Apr 02 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2024

Keywords

Comments

In an antichain of sets, no edge is a proper subset of any other.

Examples

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Links

Crossrefs

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]

A355145 Triangle read by rows: T(n,k) is the number of primitive subsets of {1,...,n} of cardinality k; n>=0, 0<=k<=ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 2, 1, 6, 7, 3, 1, 7, 12, 10, 3, 1, 8, 16, 15, 5, 1, 9, 22, 26, 13, 2, 1, 10, 28, 38, 22, 4, 1, 11, 37, 66, 60, 26, 4, 1, 12, 43, 80, 76, 35, 6, 1, 13, 54, 123, 156, 111, 41, 6, 1, 14, 64, 161, 227, 180, 74, 12
Offset: 0

Views

Author

Marcel K. Goh, Jun 20 2022

Keywords

Comments

A set is primitive if it does not contain distinct i and j such that i divides j.
For n >= 2, the alternating row sums equal -1.

Examples

			Triangle T(n,k) begins:
   n/k 0  1  2  3  4  5  6  7  8  9 10 11 12
    0  1
    1  1  1
    2  1  2
    3  1  3  1
    4  1  4  2
    5  1  5  5  2
    6  1  6  7  3
    7  1  7 12 10  3
    8  1  8 16 15  5
    9  1  9 22 26 13  2
   10  1 10 28 38 22  4
   11  1 11 37 66 60 26  4
   12  1 12 43 80 76 35  6
   ...
For n=6 and k=3 the T(6,3) = 3 primitive sets are {2,3,5}, {3,4,5}, and {4,5,6}.

Links

Crossrefs

Columns k=0..2 give: A000012, A000027, A161664.
Row sums give A051026.
T(2n,n) gives A174094.
T(2n-1,n) gives A192298 for n>=1.
Cf. A087077, A087086.

Formula

Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - Alois P. Heinz, Jun 24 2022
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