A029931 -id:A029931 - cp's OEIS frontend

A372683 Least squarefree number >= 2^n.

1, 2, 5, 10, 17, 33, 65, 129, 257, 514, 1027, 2049, 4097, 8193, 16385, 32770, 65537, 131073, 262145, 524289, 1048577, 2097154, 4194305, 8388609, 16777217, 33554433, 67108865, 134217730, 268435457, 536870913, 1073741826, 2147483649, 4294967297, 8589934594

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The terms together with their binary expansions and binary indices begin:
       1:                    1 ~ {1}
       2:                   10 ~ {2}
       5:                  101 ~ {1,3}
      10:                 1010 ~ {2,4}
      17:                10001 ~ {1,5}
      33:               100001 ~ {1,6}
      65:              1000001 ~ {1,7}
     129:             10000001 ~ {1,8}
     257:            100000001 ~ {1,9}
     514:           1000000010 ~ {2,10}
    1027:          10000000011 ~ {1,2,11}
    2049:         100000000001 ~ {1,12}
    4097:        1000000000001 ~ {1,13}
    8193:       10000000000001 ~ {1,14}
   16385:      100000000000001 ~ {1,15}
   32770:     1000000000000010 ~ {2,16}
   65537:    10000000000000001 ~ {1,17}
  131073:   100000000000000001 ~ {1,18}
  262145:  1000000000000000001 ~ {1,19}
  524289: 10000000000000000001 ~ {1,20}

For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.
Counting zeros instead of all bits gives A372473, firsts of A372472.
These are squarefree numbers at indices A372540, firsts of A372475.
Counting ones instead of all bits gives A372541, firsts of A372433.
The opposite (greatest squarefree number <= 2^n) is A372889.
The difference from 2^n is A373125.
For prime instead of squarefree we have:
- bits A372684, firsts of A035100
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes (exclusive).
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]

a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ Michel Marcus, May 29 2024

from itertools import count
from sympy import factorint
def A372683(n): return next(i for i in count(1<Chai Wah Wu, Aug 26 2024

a(n) = A005117(A372540(n)).

a(n) = A067535(2^n). - R. J. Mathar, May 31 2024

A372684 Least k such that prime(k) >= 2^n.

Original entry on oeis.org

1, 3, 5, 7, 12, 19, 32, 55, 98, 173, 310, 565, 1029, 1901, 3513, 6543, 12252, 23001, 43391, 82026, 155612, 295948, 564164, 1077872, 2063690, 3957810, 7603554, 14630844, 28192751, 54400029, 105097566, 203280222, 393615807, 762939112, 1480206280, 2874398516, 5586502349

Offset: 1

Gus Wiseman, May 30 2024

nonn

			The numbers prime(a(n)) together with their binary expansions and binary indices begin:
        2:                       10 ~ {2}
        5:                      101 ~ {1,3}
       11:                     1011 ~ {1,2,4}
       17:                    10001 ~ {1,5}
       37:                   100101 ~ {1,3,6}
       67:                  1000011 ~ {1,2,7}
      131:                 10000011 ~ {1,2,8}
      257:                100000001 ~ {1,9}
      521:               1000001001 ~ {1,4,10}
     1031:              10000000111 ~ {1,2,3,11}
     2053:             100000000101 ~ {1,3,12}
     4099:            1000000000011 ~ {1,2,13}
     8209:           10000000010001 ~ {1,5,14}
    16411:          100000000011011 ~ {1,2,4,5,15}
    32771:         1000000000000011 ~ {1,2,16}
    65537:        10000000000000001 ~ {1,17}
   131101:       100000000000011101 ~ {1,3,4,5,18}
   262147:      1000000000000000011 ~ {1,2,19}
   524309:     10000000000000010101 ~ {1,3,5,20}
  1048583:    100000000000000000111 ~ {1,2,3,21}
  2097169:   1000000000000000010001 ~ {1,5,22}
  4194319:  10000000000000000001111 ~ {1,2,3,4,23}
  8388617: 100000000000000000001001 ~ {1,4,24}

The opposite (greatest k such that prime(k) <= 2^n) is A007053.
Positions of first appearances in A035100.
The distance from prime(a(n)) to 2^n is A092131.
Counting zeros instead of all bits gives A372474, firsts of A035103.
Counting ones instead of all bits gives A372517, firsts of A014499.
For primes between powers of 2:
- sum A293697
- length A036378
- min A104080 or A014210
- max A014234, delta A013603
For squarefree numbers between powers of 2:
- sum A373123
- length A077643, run-lengths of A372475
- min A372683, delta A373125, indices A372540
- max A372889, delta A373126, indices A143658
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925, opposite A112926
Cf. A000120, A029837, A029931, A030190, A049095, A069010, A070939, A077641, A080791, A211997.

Table[PrimePi[If[n==1,2,NextPrime[2^n]]],{n,30}]

a(n) = primepi(nextprime(2^n)); \\ Michel Marcus, May 31 2024

a(n>1) = A007053(n) + 1.
a(n) = A000720(A104080(n)).
prime(a(n)) = A104080(n).
prime(a(n)) - 2^n = A092131(n).

More terms from Michel Marcus, May 31 2024

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 02 2020

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]

a(n) = A299701(A333219(n)).

A359042 Sum of partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 4, 7, 6, 9, 5, 8, 7, 10, 5, 9, 8, 12, 7, 11, 10, 14, 6, 10, 9, 13, 8, 12, 11, 15, 6, 11, 10, 15, 9, 14, 13, 18, 8, 13, 12, 17, 11, 16, 15, 20, 7, 12, 11, 16, 10, 15, 14, 19, 9, 14, 13, 18, 12, 17, 16, 21, 7, 13, 12, 18, 11, 17, 16, 22

Offset: 0

Gus Wiseman, Dec 20 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 29th composition in standard order is (1,1,2,1), with partial sums (1,2,4,5), with sum 12, so a(29) = 12.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Each n appears A000009(n) times.
The reverse version is A029931.
Comps counted by this statistic are A053632, ptns A264034, rev ptns A358194.
This is the sum of partial sums of rows of A066099.
The version for Heinz numbers of partitions is A318283, row sums of A358136.
Row sums of A358134.
A011782 counts compositions.
A065120 gives first part of standard compositions, last A001511.
A242628 lists adjusted partial sums, ranked by A253565, row sums A359043.
A358135 gives last minus first of standard compositions.
Cf. A000120, A029837, A070939, A133494, A253566, A358133, A358137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[Accumulate[stc[n]]],{n,0,100}]

A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 6, 4, 5, 4, 4, 5, 8, 7, 9, 6, 8, 7, 8, 5, 7, 6, 7, 5, 6, 5, 5, 6, 10, 9, 12, 8, 11, 10, 12, 7, 10, 9, 11, 8, 10, 9, 10, 6, 9, 8, 10, 7, 9, 8, 9, 6, 8, 7, 8, 6, 7, 6, 6, 7, 12, 11, 15, 10, 14, 13, 16, 9, 13, 12, 15, 11, 14, 13

Offset: 0

Gus Wiseman, Dec 21 2022

nonn

We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The unadjusted reverse version is A029931, row sums of A048793.
The reverse version is A161511, row sums of A125106.
Row sums of A242628, ranked by A253565.
The unadjusted version is A359042, row sums of A358134.
A011782 counts compositions.
A066099 lists standard compositions.
A358135 gives last minus first of standard compositions.
A358194 counts partitions by sum and weighted sum.
Cf. A000120, A005940, A019565, A029837, A059893, A253566, A358133, A358170.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[Accumulate[stc[n]-1]+1],{n,0,100}]

A372436 Numbers whose binary indices and prime indices have the same maximum.

Original entry on oeis.org

3, 5, 14, 22, 39, 52, 68, 85, 102, 119, 133, 152, 171, 190, 209, 228, 247, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 928, 957, 986, 1015, 1054, 1085, 1116, 1178, 1209, 1240, 1302

Offset: 1

Gus Wiseman, May 04 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.
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.
Note that a number's binary and prime indices cannot have the same minimum; see A372437.

			The binary indices of 345 are {1,4,5,7,9}, and the prime indices are {2,3,9}. Both have maximum 9, so 345 is in the sequence.
The terms together with their prime indices begin:
     3: {2}
     5: {3}
    14: {1,4}
    22: {1,5}
    39: {2,6}
    52: {1,1,6}
    68: {1,1,7}
    85: {3,7}
   102: {1,2,7}
   119: {4,7}
   133: {4,8}
   152: {1,1,1,8}
   171: {2,2,8}
The terms together with their binary expansions and binary indices begin:
     3:           11 ~ {1,2}
     5:          101 ~ {1,3}
    14:         1110 ~ {2,3,4}
    22:        10110 ~ {2,3,5}
    39:       100111 ~ {1,2,3,6}
    52:       110100 ~ {3,5,6}
    68:      1000100 ~ {3,7}
    85:      1010101 ~ {1,3,5,7}
   102:      1100110 ~ {2,3,6,7}
   119:      1110111 ~ {1,2,3,5,6,7}
   133:     10000101 ~ {1,3,8}
   152:     10011000 ~ {4,5,8}
   171:     10101011 ~ {1,2,4,6,8}

For length instead of maximum we have A071814.
For sum instead of maximum we have A372427.
Positions of zeros in A372442, for minimum instead of maximum A372437.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A030101, A066099, A096111, A304818, A355536, A359401, A359402, A372428-A372433, A372441.

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],Max[prix[#]]==Max[bix[#]]&]

A070939(a(n)) = A061395(a(n)).

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 5, 5, 4, 1, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 5, 1, 3, 3, 5, 2, 6, 6, 7, 3, 6, 3, 8, 6, 7, 8, 9, 3, 5, 6, 8, 6, 8, 7, 11, 5, 8, 8, 11, 7, 11, 9, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 9, 5, 9, 9, 9, 3, 6, 5, 9, 5, 7, 8, 11, 6, 9, 8, 11, 9, 11, 11, 11, 3, 5, 6, 8, 5, 9

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 24 2015

			0.3684125359314336523213165973278510150142413039288199683036158...

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

Jonathan M. Borwein and Peter B. Borwein, Strange series and high precision fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640; alternative link.
Martin Klazar, What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, The integration of q-series.

Cf. A000120, A010815, A029931, A242168, A258229, A258230, A258408.

Cf. A258406 (m=2), A258407 (m=3), A258404 (m=4), A258405 (m=5).

evalf(8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6)/(2*cosh(sqrt(23)*Pi/3)-1), 123);
evalf(Sum((-1)^n/((3*n-1)*n/2 + 1), n=-infinity..infinity), 123);

RealDigits[N[8*Sqrt[3/23]*Pi*Sinh[Sqrt[23]*Pi/6] / (2*Cosh[Sqrt[23]*Pi/3]-1),120]][[1]]

8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1) \\ Michel Marcus, Nov 28 2018

Equals 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1).
From Amiram Eldar, Feb 04 2024: (Start)
Equals 2 * Sum_{k=-oo..oo} (-1)^k/(3*k^2 + k + 2).
Equals Sum_{k>=0} (-1)^A000120(k)/(A029931(k)+1) (Borwein and Borwein, 1992). (End)

A326782 Numbers whose binary indices are prime numbers.

Original entry on oeis.org

0, 2, 4, 6, 16, 18, 20, 22, 64, 66, 68, 70, 80, 82, 84, 86, 1024, 1026, 1028, 1030, 1040, 1042, 1044, 1046, 1088, 1090, 1092, 1094, 1104, 1106, 1108, 1110, 4096, 4098, 4100, 4102, 4112, 4114, 4116, 4118, 4160, 4162, 4164, 4166, 4176, 4178, 4180, 4182, 5120

Offset: 1

Gus Wiseman, Jul 25 2019

nonn

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.

Write n = 2^e_1 + 2^e_2 + 2^e_3 + ..., with e_1>e_2>e_3>... We require that all the numbers e_i + 1 are primes. So 6 = 2^2+2^1 is OK because 2+1 and 1+1 are primes. 0 is OK because there are no e_i. - N. J. A. Sloane, Jul 27 2019

			The sequence of terms together with their binary indices begins:
     0: {}
     2: {2}
     4: {3}
     6: {2,3}
    16: {5}
    18: {2,5}
    20: {3,5}
    22: {2,3,5}
    64: {7}
    66: {2,7}
    68: {3,7}
    70: {2,3,7}
    80: {5,7}
    82: {2,5,7}
    84: {3,5,7}
    86: {2,3,5,7}
  1024: {11}
  1026: {2,11}
  1028: {3,11}
  1030: {2,3,11}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A029931, A048793, A070939, A326031, A326701, A326781, A326788.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(L[i]*2^(ithprime(i)-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 26 2019

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],And@@PrimeQ/@bpe[#]&]

A326853 BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105

Offset: 1

Gus Wiseman, Aug 18 2019

nonn

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence gives all BII-numbers (defined below) of set-systems that are cointersecting, meaning their dual is pairwise intersecting.

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 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.

			The sequence of all cointersecting set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}

BII-numbers of pairwise intersecting set-systems are A326910.
Cointersecting set-systems are A327039, with covering version A327040.
The T_0 or costrict case is A327052.
Cf. A029931, A048793, A326031, A327020, A327037, A327041, A327053, A327057.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]

cp's OEIS Frontend

A372683 Least squarefree number >= 2^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A372684 Least k such that prime(k) >= 2^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A359042 Sum of partial sums of the n-th composition in standard order (A066099).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A372436 Numbers whose binary indices and prime indices have the same maximum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Views

Author

Keywords

Examples

References