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.

Showing 1-4 of 4 results.

A329661 BII-number of the set-system whose MM-number is A329629(n).

Original entry on oeis.org

0, 1, 2, 8, 4, 3, 128, 16, 32768, 9, 5, 2147483648, 256, 32, 129, 10, 9223372036854775808, 6, 170141183460469231731687303715884105728, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 130, 17, 32769, 4294967296
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2019

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. 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.
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}}.

Examples

			The sequence of all set-systems together with their MM-numbers and BII-numbers begins:
             {}:  1 ~ 0
          {{1}}:  3 ~ 1
          {{2}}:  5 ~ 2
          {{3}}: 11 ~ 8
        {{1,2}}: 13 ~ 4
      {{1},{2}}: 15 ~ 3
          {{4}}: 17 ~ 128
        {{1,3}}: 29 ~ 16
          {{5}}: 31 ~ 32768
      {{1},{3}}: 33 ~ 9
    {{1},{1,2}}: 39 ~ 5
          {{6}}: 41 ~ 2147483648
        {{1,4}}: 43 ~ 256
        {{2,3}}: 47 ~ 32
      {{1},{4}}: 51 ~ 129
      {{2},{3}}: 55 ~ 10
          {{7}}: 59 ~ 9223372036854775808
    {{2},{1,2}}: 65 ~ 6
          {{8}}: 67 ~ 170141183460469231731687303715884105728
        {{2,4}}: 73 ~ 512

Crossrefs

MM-numbers of set-systems are A329629.
Cf. A000120, A005117, A048793, A056239, A070939, A112798, A302242, A302494, A326031, A329557.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
Classes of BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326752 (hypertrees), A326754 (covers).

Programs

  • Mathematica
    fbi[q_]:=If[q=={},0,Total[2^q]/2];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
das=Select[Range[100],OddQ[#]&&SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&];
Table[fbi[fbi/@primeMS/@primeMS[n]],{n,das}]

Formula

A326031(a(n)) = A302242(A329629(n)).

A329557 Smallest MM-number of a set of n nonempty sets.

Original entry on oeis.org

1, 3, 15, 165, 2145, 36465, 1057485, 32782035, 1344063435, 57794727705, 2716352202135, 160264779925965, 10737740255039655, 783855038617894815, 61924548050813690385, 5139737488217536301955, 519113486309971166497455, 56583370007786857148222595, 6393920810879914857749153235
Offset: 0

Views

Author

Gus Wiseman, Nov 17 2019

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}}.

Examples

			The sequence of terms together with their corresponding systems begins:
        1: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

Crossrefs

MM-numbers of sets of sets are A302494.
MM-numbers of sets of nonempty sets are A329629.
The version allowing empty sets is A329558.
The version without singletons is A329554.
Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]
  • PARI
    a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ Jinyuan Wang, Feb 23 2025

Formula

a(n) = A329558(n + 1)/2.

Extensions

More terms from Jinyuan Wang, Feb 23 2025

A329630 Products of distinct primes of nonprime squarefree index.

Original entry on oeis.org

1, 2, 13, 26, 29, 43, 47, 58, 73, 79, 86, 94, 101, 113, 137, 139, 146, 149, 158, 163, 167, 181, 199, 202, 226, 233, 257, 269, 271, 274, 278, 293, 298, 313, 317, 326, 334, 347, 349, 362, 373, 377, 389, 397, 398, 421, 439, 443, 449, 466, 467, 487, 491, 499, 514
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2019

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}}. This sequence lists all MM-numbers of sets of non-singleton sets.

Examples

			The sequence of terms together with their corresponding sets of sets begins:
    1: {}
    2: {{}}
   13: {{1,2}}
   26: {{},{1,2}}
   29: {{1,3}}
   43: {{1,4}}
   47: {{2,3}}
   58: {{},{1,3}}
   73: {{2,4}}
   79: {{1,5}}
   86: {{},{1,4}}
   94: {{},{2,3}}
  101: {{1,6}}
  113: {{1,2,3}}
  137: {{2,5}}
  139: {{1,7}}
  146: {{},{2,4}}
  149: {{3,4}}
  158: {{},{1,5}}
  163: {{1,8}}

Crossrefs

MM-numbers of sets of nonempty sets are A329629.
Products of primes of nonprime squarefree index are A320630.
Products of prime numbers of squarefree index are A302478.
Products of primes of nonprime index are A320628.
Cf. A005117, A056239, A112798, A302242, A302494, A329557.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&!MemberQ[primeMS[#],_?PrimeQ]&]

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}]
Showing 1-4 of 4 results.

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