A356930 -id:A356930 - cp's OEIS frontend

A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

1, 3, 5, 9, 11, 15, 17, 19, 25, 27, 31, 33, 37, 41, 45, 51, 55, 57, 59, 61, 67, 71, 75, 81, 83, 85, 93, 95, 99, 103, 107, 109, 111, 113, 121, 123, 125, 127, 131, 135, 153, 155, 157, 165, 171, 177, 179, 181, 183, 185, 187, 191, 193, 197, 201, 205, 209, 211, 213

Offset: 1

Gus Wiseman, Sep 12 2022

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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(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}}.

			The initial terms and corresponding multiset partitions:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  45: {{1},{1},{2}}
  51: {{1},{4}}
  55: {{2},{3}}
  57: {{1},{1,1,1}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Odd-size multisets are ctd by A000302, A027193, A058695, rkd by A026424.
Other types: A050330, A356932, A356933, A356934.
Other conditions: A302478, A302492, A356930, A356939, A356940, A356944, A356945.
Cf. A000040, A000720, A003963, A270995, A302242, A304969, A349050, A349055.

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

A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 36, 39, 40, 41, 44, 45, 47, 48, 50, 51, 52, 54, 55, 59, 60, 62, 64, 65, 66, 67, 68, 72, 75, 78, 80, 81, 82, 83, 85, 88, 90, 93, 94, 96, 99, 100, 102, 104, 108

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(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}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

The initial version is A356940.
Intervals are counted by A000012, A001227, ranked by A073485.
Other types: A107742, A356936, A356937, A356938.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Select[Range[100],And@@chQ/@primeMS/@primeMS[#]&]

A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(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}}.

			The initial terms and corresponding multiset partitions:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Gapless multisets are counted by A034296, ranked by A073491.
The initial version is A356955.
Other types: A356233, A356941, A356942, A356943.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A011782 counts multisets covering an initial interval.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A076610, A270995, A302242, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[100],And@@nogapQ/@primeMS/@primeMS[#]&]

A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 96, 104, 108, 113, 117, 128, 144, 156, 162, 169, 192, 208, 216, 226, 234, 243, 256, 288, 312, 324, 338, 339, 351, 384, 416, 432, 452, 468, 486, 507, 512, 576, 624, 648

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(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}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  32: {{},{},{},{},{}}
  36: {{},{},{1},{1}}
  39: {{1},{1,2}}
  48: {{},{},{},{},{1}}
  52: {{},{},{1,2}}
  54: {{},{1},{1},{1}}
  64: {{},{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

This is the initial version of A356939.
Initial intervals are counted by A010054, ranked by A002110.
Other types: A007294, A322585.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A302242, A304969.

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

A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 26, 27, 28, 32, 36, 37, 38, 39, 42, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 72, 74, 76, 78, 81, 84, 89, 91, 96, 98, 104, 106, 108, 111, 112, 113, 114, 117, 122, 126, 128, 131, 133, 144, 147, 148, 151, 152

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(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}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  32: {{},{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Multisets covering an initial interval are ctd by A011782, rkd by A055932.
This is the initial version of A356944.
Other types: A034691, A089259, A356945, A356954.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],And@@normQ/@primeMS/@primeMS[#]&]

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 4, 0, 2, 1, 0, 2, 0, 0, 0, 0, 0, 1, 7, 0, 2, 0, 0, 0, 0, 0, 7, 1, 0, 0, 4, 0, 2, 1, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 11, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 12, 0, 2, 1, 0, 2, 0

Offset: 1

Gus Wiseman, Sep 08 2022

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 a(440) = 21 multiset partitions of {1,1,1,3,5}:
  {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
                   {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
                   {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
                   {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
                                  {11}{3}{15}  {13}{115}
                                  {1}{3}{115}  {3}{1115}
                                  {1}{5}{113}  {5}{1113}
                                  {3}{111}{5}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A324929, complement A066208.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Other types: A000009, A000012, A000041, A089259, A356930.
Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.
Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],And@@(OddQ[Times@@primeMS[#]]&/@#)&]],{n,100}]

a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).

cp's OEIS Frontend

A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

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A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

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A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

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A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

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A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

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A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

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