A324739 -id:A324739

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A324738

Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

+10
13

1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784

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

OFFSET

0,2

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

Andrew Howroyd, Table of n, a(n) for n = 0..100

EXAMPLE

The a(0) = 1 through a(6) = 26 subsets:

{} {} {} {} {} {} {}

{1} {1} {1} {1} {1} {1}

{2} {2} {2} {2} {2}

{3} {3} {3} {3}

{1,3} {4} {4} {4}

{1,3} {5} {5}

{2,4} {1,3} {6}

{3,4} {1,5} {1,3}

{2,4} {1,5}

{2,5} {1,6}

{3,4} {2,4}

{4,5} {2,5}

{2,4,5} {2,6}

{3,4}

{3,6}

{4,5}

{4,6}

{5,6}

{1,3,6}

{1,5,6}

{2,4,5}

{2,4,6}

{2,5,6}

{3,4,6}

{4,5,6}

{2,4,5,6}

MATHEMATICA

Table[Length[Select[Subsets[Range[n]], !MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@FactorInteger[k]]]&]], {n, 0, 10}]

PROG

(PARI)

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}

a(n)={my(p=vector(n, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));

((k, b)->if(k>#p, 1, my(t=self()(k+1, b)); if(bitnegimply(p[k], b), t+=if(bittest(d, k), self()(k+1, b+(1<<k)), t)); t))(1, 0)} \\ Andrew Howroyd, Aug 16 2019

CROSSREFS

The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.

Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.

Cf. A324695, A324736, A324741, A324750, A324755, A324760.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 13 2019

EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

STATUS

approved

A324755

Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

+10
9

1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 10, 7, 16, 14, 23, 23, 35, 34, 53, 54, 75, 80, 112, 115, 160, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011

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

OFFSET

0,5

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.

For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.

LINKS

Table of n, a(n) for n=0..53.

EXAMPLE

The a(2) = 1 through a(10) = 10 integer partitions (A = 10):

(2) (3) (4) (5) (6) (7) (8) (9) (A)

(22) (33) (43) (44) (54) (55)

(42) (52) (62) (63) (64)

(222) (422) (72) (73)

(2222) (333) (82)

(522) (433)

(442)

(622)

(4222)

(22222)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], !MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@If[k==1, {}, FactorInteger[k]]]]&]], {n, 0, 30}]

CROSSREFS

The subset version is A324739, with maximal case A324762. The strict case is A324750. The Heinz number version is A324760. An infinite version is A324694.

Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 16 2019

STATUS

approved

A324750

Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

+10
8

1, 0, 1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

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

OFFSET

0,7

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=0..59.

EXAMPLE

The a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):

2 3 4 5 6 7 8 9 A B C D E F G H

42 43 62 54 64 65 75 76 86 87 97 98

52 63 73 83 84 85 95 96 A6 A7

72 82 542 93 94 A4 A5 C4 B6

A2 A3 B3 B4 D3 C5

642 B2 C2 C3 E2 D4

643 752 D2 763 E3

652 842 654 862 F2

762 943 854

843 A42 863

852 872

A43

A52

B42

6542

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!MemberQ[#, 1]&&!MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@FactorInteger[k]]]&]], {n, 0, 30}]

CROSSREFS

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.

Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.

Cf. A324696, A324737, A324742, A324744, A324764.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 15 2019

STATUS

approved

A324760

Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

+10
8

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139

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

LINKS

Table of n, a(n) for n=1..62.

EXAMPLE

The sequence of terms together with their prime indices begins:

1: {}

3: {2}

5: {3}

7: {4}

9: {2,2}

11: {5}

13: {6}

17: {7}

19: {8}

21: {2,4}

23: {9}

25: {3,3}

27: {2,2,2}

29: {10}

31: {11}

33: {2,5}

35: {3,4}

37: {12}

39: {2,6}

41: {13}

MATHEMATICA

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

Select[Range[100], !MemberQ[primeMS[#], k_/; SubsetQ[primeMS[#], primeMS[k]]]&]

CROSSREFS

The subset version is A324739, with maximal case A324762. The strict integer partition version is A324750. The integer partition version is A324755. An infinite version is A324694.

Cf. A000720, A001221, A007097, A056239, A112798, A289509, A290822, A306844, A324695, A324696, A324737, A324744.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 17 2019

STATUS

approved

A324762

Number of maximal subsets of {2...n} containing no element whose prime indices all belong to the subset.

+10
8

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 40, 40, 52, 52, 64, 64, 72, 72, 144, 144, 176, 176, 200, 200, 232, 232, 464, 464, 464, 464, 536, 536, 1072, 1072, 1072, 1072, 2144, 2144, 2400, 2400, 2400, 2400, 4800, 4800, 4800, 4800, 4800

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

OFFSET

1,3

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

Andrew Howroyd, Table of n, a(n) for n = 1..100

EXAMPLE

The a(2) = 1 through a(9) = 6 maximal subsets:

{2} {2} {2,4} {3,4} {3,4,6} {3,4,6} {3,4,6,8} {2,4,5,6,8}

{3} {3,4} {2,4,5} {2,4,5,6} {3,6,7} {3,6,7,8} {2,5,6,7,8}

{2,4,5,6} {2,4,5,6,8} {3,4,6,8,9}

{2,5,6,7} {2,5,6,7,8} {3,6,7,8,9}

{4,5,6,8,9}

{5,6,7,8,9}

MATHEMATICA

maxim[s_]:=Complement[s, Last/@Select[Tuples[s, 2], UnsameQ@@#&&SubsetQ@@#&]];

Table[Length[maxim[Select[Subsets[Range[2, n]], !MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@FactorInteger[k]]]&]]], {n, 10}]

PROG

(PARI)

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}

a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));

my(ismax(b)=for(k=1, #p, if(!bittest(b, k) && bitnegimply(p[k], b), my(e=bitor(b, 1<<k), f=0); for(j=k+1, #p, if(bittest(b, j) && !bitnegimply(p[j], e), f=1; break)); if(!f, return(0)) )); 1);

my(recurse(k, b)=if(k>#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<<k)))));

recurse(1, 0)} \\ Andrew Howroyd, Aug 27 2019

CROSSREFS

The non-maximal version is A324739.

The version for subsets of {1...n} is A324744.

An infinite version is A324694.

Cf. A076078, A084422, A085945, A112798, A276625, A304360, A306844.

Cf. A324695, A324738, A324742, A324743, A324763, A324750, A324754, A324755, A324760.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 17 2019

EXTENSIONS

Terms a(16) and beyond from Andrew Howroyd, Aug 27 2019

STATUS

approved

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