A305052 -id:A305052

A305078

Heinz numbers of connected integer partitions.

+10
85

2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

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

OFFSET

1,1

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence lists all Heinz numbers of multisets S such that G(S) is a connected graph.

LINKS

Table of n, a(n) for n=1..60.

Madeline Locus Dawsey, Tyler Russell and Dannie Urban, Polynomials Associated to Integer Partitions, arXiv:2108.00943 [math.NT], 2021.

EXAMPLE

The sequence of all connected multiset multisystems (see A302242, A112798) begins:

2: {{}}

3: {{1}}

5: {{2}}

7: {{1,1}}

9: {{1},{1}}

11: {{3}}

13: {{1,2}}

17: {{4}}

19: {{1,1,1}}

21: {{1},{1,1}}

23: {{2,2}}

25: {{2},{2}}

27: {{1},{1},{1}}

29: {{1,3}}

31: {{5}}

37: {{1,1,2}}

39: {{1},{1,2}}

41: {{6}}

43: {{1,4}}

47: {{2,3}}

49: {{1,1},{1,1}}

53: {{1,1,1,1}}

57: {{1},{1,1,1}}

59: {{7}}

61: {{1,2,2}}

63: {{1},{1},{1,1}}

65: {{2},{1,2}}

67: {{8}}

71: {{1,1,3}}

73: {{2,4}}

79: {{1,5}}

81: {{1},{1},{1},{1}}

83: {{9}}

87: {{1},{1,3}}

89: {{1,1,1,2}}

91: {{1,1},{1,2}}

97: {{3,3}}

MATHEMATICA

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

Select[Range[300], Length[zsm[primeMS[#]]]==1&]

CROSSREFS

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 24 2018

STATUS

approved

A305079

Number of connected components of the integer partition with Heinz number n.

+10
63

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 5, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 4, 1, 2, 1, 6, 1, 3, 1, 3, 2, 3, 1, 4, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 1, 3, 2, 2, 1

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

OFFSET

1,4

COMMENTS

First differs from |A305052(n)| at a(169) = 1, A305052(169) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. If S is the integer partition with Heinz number n, a(n) is the number of connected components of G(S).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

FORMULA

For all n, k > 0, we have a(2^n * k) = n + a(k).

For all x, y > 0, we have a(x * y) <= a(x) + a(y).

For x, y > 0 strongly coprime, we have a(x * y) = a(x) + a(y). Strongly coprime means every prime index of x is coprime to every prime index of y, where a prime index of n is a number m such that prime(m) divides n.

a(n) = A305501(A064989(n)) + A007814(n). - Antti Karttunen, Nov 10 2018

EXAMPLE

The a(315) = 2 connected components of {2,2,3,4} are {{3},{2,2,4}}.

MATHEMATICA

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

Table[Length[zsm[primeMS[n]]], {n, 100}]

PROG

(PARI)

zero_first_elem_and_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(1!=gcd(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };

A007814(n) = valuation(n, 2);

A000265(n) = (n/2^A007814(n));

A305079(n) = if(!(n%2), A007814(n)+A305079(A000265(n)), my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Nov 10 2018

CROSSREFS

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305078, A305501.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 24 2018

EXTENSIONS

Terms and Mathematica program corrected by Gus Wiseman, Nov 10 2018

STATUS

approved

A320456

Numbers whose multiset multisystem spans an initial interval of positive integers.

+10
40

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117

(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. The n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

2: {{}}

3: {{1}}

4: {{},{}}

6: {{},{1}}

7: {{1,1}}

8: {{},{},{}}

9: {{1},{1}}

12: {{},{},{1}}

13: {{1,2}}

14: {{},{1,1}}

15: {{1},{2}}

16: {{},{},{},{}}

18: {{},{1},{1}}

19: {{1,1,1}}

21: {{1},{1,1}}

24: {{},{},{},{1}}

26: {{},{1,2}}

27: {{1},{1},{1}}

28: {{},{},{1,1}}

30: {{},{1},{2}}

32: {{},{},{},{},{}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[100], normQ[primeMS/@primeMS[#]]&]

CROSSREFS

Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A320461

MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

+10
24

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691

(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. 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=1..43.

Eric Weisstein's World of Mathematics, Simple Graph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

7: {{1,1}}

13: {{1,2}}

91: {{1,1},{1,2}}

161: {{1,1},{2,2}}

299: {{2,2},{1,2}}

329: {{1,1},{2,3}}

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

611: {{1,2},{2,3}}

667: {{2,2},{1,3}}

1261: {{3,3},{1,2}}

1363: {{1,3},{2,3}}

1937: {{1,2},{3,4}}

2021: {{1,4},{2,3}}

2093: {{1,1},{2,2},{1,2}}

2117: {{1,3},{2,4}}

2639: {{1,1},{1,2},{1,3}}

4277: {{1,1},{1,2},{2,3}}

4669: {{1,1},{2,2},{1,3}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[10000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A320458

MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

+10
13

1, 13, 377, 611, 1363, 1937, 2021, 2117, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 143663, 146653, 147533, 153023, 159659, 167243, 170839, 203087, 237679, 243893, 265369, 271049, 276877, 290029, 301129, 315433, 467711

(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. 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=1..37.

Eric Weisstein's World of Mathematics, Simple Graph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

13: {{1,2}}

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

611: {{1,2},{2,3}}

1363: {{1,3},{2,3}}

1937: {{1,2},{3,4}}

2021: {{1,4},{2,3}}

2117: {{1,3},{2,4}}

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

17719: {{1,2},{1,3},{2,3}}

26273: {{1,2},{1,4},{2,3}}

27521: {{1,2},{1,3},{2,4}}

44603: {{1,2},{2,3},{2,4}}

56173: {{1,2},{1,3},{3,4}}

58609: {{1,3},{1,4},{2,3}}

83291: {{1,2},{1,4},{3,4}}

91031: {{1,3},{1,4},{2,4}}

91039: {{1,2},{2,3},{3,4}}

99499: {{1,3},{2,3},{2,4}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[10000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(And[SquareFreeQ[#], Length[primeMS[#]]==2]&/@primeMS[#])]&]

CROSSREFS

Cf. A001222, A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A302491, A305052.

Cf. A320456, A320459, A320461, A320462, A320463.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A320462

MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

+10
12

1, 7, 13, 49, 91, 161, 169, 299, 329, 343, 377, 611, 637, 667, 1127, 1183, 1261, 1363, 1937, 2021, 2093, 2117, 2197, 2303, 2401, 2639, 3703, 3887, 4277, 4459, 4669, 4901, 6877, 7567, 7889, 7943, 8281, 8671, 8827, 9541, 10933, 13559, 14053, 14147, 14651, 14819

(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. 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=1..46.

Eric Weisstein's World of Mathematics, Simple Graph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

7: {{1,1}}

13: {{1,2}}

49: {{1,1},{1,1}}

91: {{1,1},{1,2}}

161: {{1,1},{2,2}}

169: {{1,2},{1,2}}

299: {{2,2},{1,2}}

329: {{1,1},{2,3}}

343: {{1,1},{1,1},{1,1}}

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

611: {{1,2},{2,3}}

637: {{1,1},{1,1},{1,2}}

667: {{2,2},{1,3}}

1127: {{1,1},{1,1},{2,2}}

1183: {{1,1},{1,2},{1,2}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[10000], And[normQ[primeMS/@primeMS[#]], And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A305052, A305078.

Cf. A320456, A320458, A320459, A320461, A320533.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A320459

MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

+10
8

1, 13, 169, 377, 611, 1363, 1937, 2021, 2117, 2197, 4901, 7943, 10933, 16211, 17719, 25181, 26273, 27521, 28561, 28717, 39527, 44603, 56173, 58609, 61393, 63713, 64061, 83291, 86903, 91031, 91039, 94987, 99499, 103259, 141401, 142129, 143663, 146653, 147533

(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. 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=1..39.

Eric Weisstein's World of Mathematics, Simple Graph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

13: {{1,2}}

169: {{1,2},{1,2}}

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

611: {{1,2},{2,3}}

1363: {{1,3},{2,3}}

1937: {{1,2},{3,4}}

2021: {{1,4},{2,3}}

2117: {{1,3},{2,4}}

2197: {{1,2},{1,2},{1,2}}

4901: {{1,2},{1,2},{1,3}}

7943: {{1,2},{1,2},{2,3}}

10933: {{1,2},{1,3},{1,3}}

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

17719: {{1,2},{1,3},{2,3}}

25181: {{1,2},{1,2},{3,4}}

26273: {{1,2},{1,4},{2,3}}

27521: {{1,2},{1,3},{2,4}}

28561: {{1,2},{1,2},{1,2},{1,2}}

28717: {{1,2},{2,3},{2,3}}

39527: {{1,3},{1,3},{2,3}}

44603: {{1,2},{2,3},{2,4}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[100000], And[normQ[primeMS/@primeMS[#]], And@@(And[SquareFreeQ[#], Length[primeMS[#]]==2]&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052, A305078.

Cf. A320456, A320458, A320461, A320462, A320464.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved

A320532

MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

+10
8

1, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 133, 151, 161, 223, 247, 251, 259, 281, 299, 311, 329, 359, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 953, 1007, 1057, 1069, 1073, 1157

(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. 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=1..53.

Wikipedia, Hypergraph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

7: {{1,1}}

13: {{1,2}}

19: {{1,1,1}}

37: {{1,1,2}}

53: {{1,1,1,1}}

61: {{1,2,2}}

89: {{1,1,1,2}}

91: {{1,1},{1,2}}

113: {{1,2,3}}

131: {{1,1,1,1,1}}

133: {{1,1},{1,1,1}}

151: {{1,1,2,2}}

161: {{1,1},{2,2}}

223: {{1,1,1,1,2}}

247: {{1,2},{1,1,1}}

251: {{1,2,2,2}}

259: {{1,1},{1,1,2}}

281: {{1,1,2,3}}

299: {{1,2},{2,2}}

311: {{1,1,1,1,1,1}}

329: {{1,1},{2,3}}

359: {{1,1,1,2,2}}

371: {{1,1},{1,1,1,1}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[1000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320461, A320463, A320464, A320533.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 14 2018

STATUS

approved

A320533

MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

+10
8

1, 7, 13, 19, 37, 49, 53, 61, 89, 91, 113, 131, 133, 151, 161, 169, 223, 247, 251, 259, 281, 299, 311, 329, 343, 359, 361, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 637, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 931, 953

(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. 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=1..54.

Wikipedia, Hypergraph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

7: {{1,1}}

13: {{1,2}}

19: {{1,1,1}}

37: {{1,1,2}}

49: {{1,1},{1,1}}

53: {{1,1,1,1}}

61: {{1,2,2}}

89: {{1,1,1,2}}

91: {{1,1},{1,2}}

113: {{1,2,3}}

131: {{1,1,1,1,1}}

133: {{1,1},{1,1,1}}

151: {{1,1,2,2}}

161: {{1,1},{2,2}}

169: {{1,2},{1,2}}

223: {{1,1,1,1,2}}

247: {{1,2},{1,1,1}}

251: {{1,2,2,2}}

259: {{1,1},{1,1,2}}

281: {{1,1,2,3}}

299: {{1,2},{2,2}}

311: {{1,1,1,1,1,1}}

329: {{1,1},{2,3}}

343: {{1,1},{1,1},{1,1}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[1000], And[normQ[primeMS/@primeMS[#]], And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320462, A320463, A320464, A320532.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 14 2018

STATUS

approved

A320463

MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

+10
5

1, 13, 113, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 3277, 4537, 4859, 5249, 5311, 7423, 8249, 8507, 16211, 16403, 16559, 16783, 16837, 17719, 20443, 20453, 24553, 25477, 26273, 26969, 27521, 34567, 37439, 39437, 41689, 42011, 42137, 42601, 43873, 43957

(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. 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=1..41.

Wikipedia, Hypergraph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

1: {}

13: {{1,2}}

113: {{1,2,3}}

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

611: {{1,2},{2,3}}

1291: {{1,2,3,4}}

1363: {{1,3},{2,3}}

1469: {{1,2},{1,2,3}}

1937: {{1,2},{3,4}}

2021: {{1,4},{2,3}}

2117: {{1,3},{2,4}}

3277: {{1,3},{1,2,3}}

4537: {{1,2},{1,3,4}}

4859: {{1,4},{1,2,3}}

5249: {{1,3},{1,2,4}}

5311: {{2,3},{1,2,3}}

7423: {{1,2},{2,3,4}}

8249: {{2,4},{1,2,3}}

8507: {{2,3},{1,2,4}}

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

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];

Select[Range[10000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(And[SquareFreeQ[#], PrimeOmega[#]>1]&/@primeMS[#])]&]

CROSSREFS

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320458, A320464, A320532, A320533.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 13 2018

STATUS

approved