A326781 -id:A326781

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A326782

Numbers whose binary indices are prime numbers.

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

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

LINKS

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

EXAMPLE

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}

MAPLE

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

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[0, 100], And@@PrimeQ/@bpe[#]&]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 25 2019

STATUS

approved

A367770

Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.

+10
18

1, 1, 2, 15, 558, 81282, 39400122, 61313343278, 309674769204452

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.

LINKS

Table of n, a(n) for n=0..8.

Wikipedia, Axiom of choice.

EXAMPLE

The a(3) = 15 set-systems:

{}

{{1,2},{1,3}}

{{1,2},{2,3}}

{{1,2},{1,2,3}}

{{1,3},{2,3}}

{{1,3},{1,2,3}}

{{2,3},{1,2,3}}

{{1,2},{1,3},{2,3}}

{{1,2},{1,3},{1,2,3}}

{{1,2},{2,3},{1,2,3}}

{{1,3},{2,3},{1,2,3}}

MATHEMATICA

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

CROSSREFS

Set-systems without singletons are counted by A016031, covering A323816.

The version for simple graphs is A133686, covering A367869.

The complement is counted by A367769.

The complement allowing singletons and empty sets is A367901.

Allowing singletons gives A367902, ranks A367906.

The complement allowing singletons is A367903, ranks A367907.

These set-systems have ranks A367906 /\ A326781.

A000372 counts antichains, covering A006126, nonempty A014466.

A003465 counts covering set-systems, unlabeled A055621.

A058891 counts set-systems, unlabeled A000612.

A323818 counts covering connected set-systems, unlabeled A323819.

Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 05 2023

EXTENSIONS

a(6)-a(8) from Christian Sievers, Jul 28 2024

STATUS

approved

A329554

Smallest MM-number of a set of n nonempty sets with no singletons.

+10
5

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

A multiset multisystem is a finite multiset of finite multisets. 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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

LINKS

Table of n, a(n) for n=0..15.

FORMULA

a(n) = Product_{i = 1..n} prime(A120944(i)).

EXAMPLE

The sequence of terms together with their corresponding systems begins:

1: {}

13: {{1,2}}

377: {{1,2},{1,3}}

16211: {{1,2},{1,3},{1,4}}

761917: {{1,2},{1,3},{1,4},{2,3}}

MATHEMATICA

sqvs=Select[Range[2, 30], SquareFreeQ[#]&&!PrimeQ[#]&];

Table[Times@@Prime/@Take[sqvs, k], {k, 0, Length[sqvs]}]

CROSSREFS

The smallest BII-number of a set of n sets is A000225(n).

BII-numbers of set-systems with no singletons are A326781.

MM-numbers of sets of nonempty sets are the odd terms of A302494.

MM-numbers of multisets of nonempty non-singleton sets are A320629.

The version with empty edges is A329556.

The version with singletons is A329557.

The version with empty edges and singletons is A329558.

Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.

Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 17 2019

STATUS

approved

A371443

Numbers whose binary indices are nonprime numbers.

+10
4

1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

The terms together with their binary expansions and binary indices begin:

1: 1 ~ {1}

8: 1000 ~ {4}

9: 1001 ~ {1,4}

32: 100000 ~ {6}

33: 100001 ~ {1,6}

40: 101000 ~ {4,6}

41: 101001 ~ {1,4,6}

128: 10000000 ~ {8}

129: 10000001 ~ {1,8}

136: 10001000 ~ {4,8}

137: 10001001 ~ {1,4,8}

160: 10100000 ~ {6,8}

161: 10100001 ~ {1,6,8}

168: 10101000 ~ {4,6,8}

169: 10101001 ~ {1,4,6,8}

256: 100000000 ~ {9}

257: 100000001 ~ {1,9}

264: 100001000 ~ {4,9}

265: 100001001 ~ {1,4,9}

288: 100100000 ~ {6,9}

289: 100100001 ~ {1,6,9}

296: 100101000 ~ {4,6,9}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[100], And@@Not/@PrimeQ/@bpe[#]&]

CROSSREFS

For powers of 2 instead of nonprime numbers we have A253317.

For prime indices instead of binary indices we have A320628.

For prime instead of nonprime we have A326782.

For composite numbers we have A371444.

An opposite version is A371449.

A000040 lists prime numbers, complement A018252.

A000961 lists prime-powers.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A001222, A005117, A326781, A368109, A368533, A371289, A371452.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 30 2024

STATUS

approved

A371444

Numbers whose binary indices are composite numbers.

+10
4

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..49.

EXAMPLE

The terms together with their binary expansions and binary indices begin:

8: 1000 ~ {4}

32: 100000 ~ {6}

40: 101000 ~ {4,6}

128: 10000000 ~ {8}

136: 10001000 ~ {4,8}

160: 10100000 ~ {6,8}

168: 10101000 ~ {4,6,8}

256: 100000000 ~ {9}

264: 100001000 ~ {4,9}

288: 100100000 ~ {6,9}

296: 100101000 ~ {4,6,9}

384: 110000000 ~ {8,9}

392: 110001000 ~ {4,8,9}

416: 110100000 ~ {6,8,9}

424: 110101000 ~ {4,6,8,9}

512: 1000000000 ~ {10}

520: 1000001000 ~ {4,10}

544: 1000100000 ~ {6,10}

552: 1000101000 ~ {4,6,10}

640: 1010000000 ~ {8,10}

648: 1010001000 ~ {4,8,10}

672: 1010100000 ~ {6,8,10}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[100], EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

CROSSREFS

For powers of 2 instead of composite numbers we have A253317.

For prime indices we have the even case of A320628.

For prime instead of composite we have A326782.

This is the even case of A371444.

An opposite version is A371449.

A000040 lists prime numbers, complement A018252.

A000961 lists prime-powers.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 30 2024

STATUS

approved

A371449

Numbers whose prime indices are not powers of 2.

+10
3

1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

The terms together with their prime indices begin:

1: {} 85: {3,7} 169: {6,6} 253: {5,9}

5: {3} 89: {24} 173: {40} 257: {55}

11: {5} 97: {25} 179: {41} 263: {56}

13: {6} 101: {26} 181: {42} 269: {57}

17: {7} 103: {27} 185: {3,12} 271: {58}

23: {9} 107: {28} 187: {5,7} 275: {3,3,5}

25: {3,3} 109: {29} 191: {43} 277: {59}

29: {10} 113: {30} 193: {44} 281: {60}

31: {11} 115: {3,9} 197: {45} 283: {61}

37: {12} 121: {5,5} 199: {46} 289: {7,7}

41: {13} 125: {3,3,3} 205: {3,13} 293: {62}

43: {14} 127: {31} 211: {47} 295: {3,17}

47: {15} 137: {33} 215: {3,14} 299: {6,9}

55: {3,5} 139: {34} 221: {6,7} 305: {3,18}

59: {17} 143: {5,6} 223: {48} 307: {63}

61: {18} 145: {3,10} 227: {49} 313: {65}

65: {3,6} 149: {35} 229: {50} 317: {66}

67: {19} 151: {36} 233: {51} 319: {5,10}

71: {20} 155: {3,11} 235: {3,15} 325: {3,3,6}

73: {21} 157: {37} 239: {52} 331: {67}

79: {22} 163: {38} 241: {53} 335: {3,19}

83: {23} 167: {39} 251: {54} 337: {68}

MATHEMATICA

Select[Range[100], And@@Not/@IntegerQ/@Log[2, PrimePi/@First/@FactorInteger[#]]&]

CROSSREFS

Partitions of this type are counted by A101417.

For binary indices instead of prime indices we have A326781.

Requiring powers of two gives A318400, for binary indices A253317.

An opposite version is A371443.

For primes instead of powers of 2 we have A320628.

A000040 lists prime numbers, complement A018252.

A000961 lists prime-powers.

A048793 lists binary indices, reverse A272020, length A000120, sum A029931.

A057716 lists non-powers of 2.

A070939 gives length of binary expansion.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

Cf. A005117, A326782, A371451, A371444.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 31 2024

STATUS

approved

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