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A329661 BII-number of the set-system whose MM-number is A329629(n). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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}}.

LINKS

Table of n, a(n) for n=1..26.

FORMULA

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

EXAMPLE

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

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}]

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).

Sequence in context: A085993 A010595 A109594 * A254446 A197486 A086311

Adjacent sequences:  A329658 A329659 A329660 * A329663 A329664 A329665

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 19 2019

STATUS

approved

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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)