A302242 -id:A302242 - cp's OEIS frontend

A302796 Squarefree numbers whose prime indices are relatively prime. Nonprime Heinz numbers of strict integer partitions with relatively prime parts.

1, 2, 6, 10, 14, 15, 22, 26, 30, 33, 34, 35, 38, 42, 46, 51, 55, 58, 62, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 102, 105, 106, 110, 114, 118, 119, 122, 123, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165, 166, 170, 174, 177, 178, 182

Offset: 1

Gus Wiseman, Apr 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are relatively prime if they have no common divisor other than 1. A single number is not considered relatively prime unless it is equal to 1.

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

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
06 : {1,2}
10 : {1,3}
14 : {1,4}
15 : {2,3}
22 : {1,5}
26 : {1,6}
30 : {1,2,3}
33 : {2,5}
34 : {1,7}
35 : {3,4}
38 : {1,8}
42 : {1,2,4}
46 : {1,9}
51 : {2,7}
55 : {3,5}
58 : {1,10}
62 : {1,11}
66 : {1,2,5}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A302242, A302505, A302696, A302697, A302698, A302797, A302798.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&GCD@@PrimePi/@FactorInteger[#][[All,1]]===1]&]

isok(n) = {if (n == 1, return (1)); if (issquarefree(n), my(f = factor(n)); return (gcd(vector(#f~, k, primepi(f[k,1]))) == 1););} \\ Michel Marcus, Apr 13 2018

A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Dec 09 2019

nonn

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}}.

a(n) is a divisor of A303975(n)!.

			The vertex-permutations of {{1,2},{2,3,3}} are:
  {{1,2},{1,3,3}}
  {{1,2},{2,3,3}}
  {{1,3},{1,2,2}}
  {{1,3},{2,2,3}}
  {{2,3},{1,1,2}}
  {{2,3},{1,1,3}}
so a(4927) = 6.

Positions of 1's are A330232.
Positions of first appearances are A330230 and A330233.
The BII-number version is A330231.
Cf. A001055, A003238, A007716, A055621, A056239, A112798, A302242, A303975, A322847, A330194, A330218, A330223, A330227, A330236.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[graprms[primeMS/@primeMS[n]]],{n,100}]

A330944 Number of nonprime prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 13 2020

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.

			24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.

The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
Numbers whose prime indices are not all prime are A330945.
Cf. A000040, A000720, A001222, A007097, A018252, A056239, A112798, A302242, A320629, A320633, A330946, A330947.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020

a(n) + A257994(n) = A001222(n).

Additive with a(p^e) = e if primepi(p) is nonprime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A304713 Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173

Offset: 1

Gus Wiseman, May 17 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of entries together with their corresponding multiset multisystems (see A302242) begins:
1:  {}
2:  {{}}
3:  {{1}}
5:  {{2}}
7:  {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}

Cf. A000009, A005117, A006126, A056239, A073576, A285572, A285573, A293606, A293993, A302696, A302796, A303362, A303365, A304711.

Select[Range[300],SquareFreeQ[#]&&Select[Tuples[PrimePi/@First/@FactorInteger[#],2],UnsameQ@@#&&Divisible@@#&]==={}&]

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 85, 89, 91, 93, 97, 99, 101, 103, 107, 109, 113, 121, 123, 125, 127, 128, 131, 135, 137, 139, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The terms together with their corresponding multiset multisystems (A302242):
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A001222, A038041, A112798, A302242, A306017, A317583, A319066, A319169, A320325, A322794, A326533, A326534, A326535, A326536, A326537.

Select[Range[100],SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

is(n) = #Set(apply(p -> bigomega(primepi(p)), factor(n)[,1]~))<=1 \\ Rémy Sigrist, Oct 11 2018

A302243 Total weight of the n-th twice-odd-factored multiset partition.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 03 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-odd-factored multiset partition is constructed by factoring 2n + 1 into prime numbers and then factoring each prime index into prime numbers and taking their prime indices.

			Sequence of multiset partitions begins: (), ((1)), ((2)), ((11)), ((1)(1)), ((3)), ((12)), ((1)(2)), ((4)), ((111)), ((1)(11)), ((22)), ((2)(2)), ((1)(1)(1)), ((13)), ((5)), ((1)(3)), ((2)(11)), ((112)), ((1)(12)), ((6)).

Cf. A003963, A007716, A034691, A048673, A056239, A061775, A063834 A064216, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302242.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Sum[PrimeOmega[k],{k,primeMS[2n-1]}],{n,100}]

a(n) = A302242(2n + 1).

A320629 Products of odd primes of nonprime index.

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 49, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

The asymptotic density of this sequence is (1/2) * Product_{p in A006450} (1 - 1/p) = 1/(2*Sum_{n>=1} 1/A076610(n)) < 1/6. - Amiram Eldar, Feb 02 2021

			The sequence of terms begins:
    1 = 1
    7 = prime(4)
   13 = prime(6)
   19 = prime(8)
   23 = prime(9)
   29 = prime(10)
   37 = prime(12)
   43 = prime(14)
   47 = prime(15)
   49 = prime(4)^2
   53 = prime(16)
   61 = prime(18)
   71 = prime(20)
   73 = prime(21)
   79 = prime(22)
   89 = prime(24)
   91 = prime(4)*prime(6)
   97 = prime(25)
  101 = prime(26)
  103 = prime(27)
  107 = prime(28)
  113 = prime(30)
  131 = prime(32)
  133 = prime(4)*prime(8)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  151 = prime(36)
  161 = prime(4)*prime(9)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A320533, A320628, A320630, A320631, A320633.

Select[Range[1,100,2],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

A330945 Numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84, 86, 87

Offset: 1

Gus Wiseman, Jan 13 2020

nonn

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 sequence of terms together with their prime indices of prime indices begins:
   2: {{}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  29: {{1,3}}

Complement of A076610 (products of primes of prime index).
Numbers n such that A330944(n) > 0.
The restriction to odd terms is A330946.
The restriction to nonprimes is A330948.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320633, A330943, A330947, A330949.

Select[Range[100],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A326535 MM-numbers of multiset partitions where each part has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298540 in having 187.

These are numbers where each prime index has a different sum of prime indices. 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 obtained by taking the multiset of prime indices of each prime index 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}}.

			The sequence of multiset partitions where each part has a different sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}

Robert Price, Table of n, a(n) for n = 1..9715
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@primeMS/@primeMS[#]&]

A303837 Number of z-trees with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 6, 1, 1, 2, 2, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. Then a z-tree is a finite connected set of pairwise indivisible positive integers greater than 1 with clutter density -1.
This is a generalization to multiset systems of the usual definition of hypertree (viz. connected hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A030019(k).

			The a(72) = 6 z-trees together with the corresponding multiset systems (see A112798, A302242) are the following.
      (72): {{1,1,1,2,2}}
    (8,18): {{1,1,1},{1,2,2}}
    (8,36): {{1,1,1},{1,1,2,2}}
    (9,24): {{2,2},{1,1,1,2}}
   (6,8,9): {{1,2},{1,1,1},{2,2}}
  (8,9,12): {{1,1,1},{2,2},{1,1,2}}
The a(60) = 10 z-trees together with the corresponding multiset systems are the following.
       (60): {{1,1,2,3}}
     (4,30): {{1,1},{1,2,3}}
     (6,20): {{1,2},{1,1,3}}
    (10,12): {{1,3},{1,1,2}}
    (12,15): {{1,1,2},{2,3}}
    (12,20): {{1,1,2},{1,1,3}}
    (15,20): {{2,3},{1,1,3}}
   (4,6,10): {{1,1},{1,2},{1,3}}
   (4,6,15): {{1,1},{1,2},{2,3}}
  (4,10,15): {{1,1},{1,3},{2,3}}

Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A006126, A030019, A048143, A076078, A112798, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303838, A304118.

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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zensity[#]==-1,zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,2,50}]

cp's OEIS Frontend

A302796 Squarefree numbers whose prime indices are relatively prime. Nonprime Heinz numbers of strict integer partitions with relatively prime parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A330944 Number of nonprime prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A304713 Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A320324 Numbers of which each prime index has the same number of prime factors, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A302243 Total weight of the n-th twice-odd-factored multiset partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A320629 Products of odd primes of nonprime index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A330945 Numbers whose prime indices are not all prime numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica