A367107 -id:A367107 - cp's OEIS frontend

A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 30, 32, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 56, 57, 60, 64, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 110, 111, 112, 114, 120, 123, 125, 126, 128, 129, 130, 132, 135, 138, 140, 141

Offset: 1

Gus Wiseman, Nov 14 2023

nonn

Also numbers m whose prime indices have a submultiset summing to bigomega(m).
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.
These are the Heinz numbers of the partitions counted by A367212.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224*   A367225    A367226    A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict integer partitions, counted by A000009.
A066208 ranks partitions into odd parts, also counted by A000009.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A229816 counts partitions whose length is not a part, ranks A367107.
A237668 counts sum-full partitions, ranks A364532.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A299729, A304792, A363225, A365830.

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

A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 58, 59, 61, 62, 63, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 113

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

Also numbers m whose prime indices do not have a submultiset summing to bigomega(m).
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.
These are the Heinz numbers of the partitions counted by A367213.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        29: {10}       58: {1,10}
     5: {3}        31: {11}       59: {17}
     7: {4}        34: {1,7}      61: {18}
    10: {1,3}      35: {3,4}      62: {1,11}
    11: {5}        37: {12}       63: {2,2,4}
    13: {6}        38: {1,8}      65: {3,6}
    14: {1,4}      41: {13}       67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    22: {1,5}      46: {1,9}      73: {21}
    23: {9}        47: {15}       74: {1,12}
    25: {3,3}      49: {4,4}      76: {1,1,8}
    26: {1,6}      52: {1,1,6}    77: {4,5}
    27: {2,2,2}    53: {16}       79: {22}
    28: {1,1,4}    55: {3,5}      82: {1,13}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225*   A367226    A367227
A000700 counts self-conjugate partitions, ranks A088902.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A229816 counts partitions whose length is not a part, ranks A367107.
A237667 counts sum-free partitions, ranks A364531.
A365924 counts incomplete partitions, ranks A365830.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A288728, A304792, A325761, A325781, A364345, A364347.

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

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104

Offset: 1

Gus Wiseman, Nov 15 2023

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.

These are the Heinz numbers of the partitions counted by A367218.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226*   A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict partitions, counted by A000009.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A066208 ranks partitions into odd parts, counted by A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A237668 counts sum-full partitions, ranks A364532.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]

A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 63, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 117, 119, 121, 127, 131, 133, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157, 161, 163

Offset: 1

Gus Wiseman, Nov 15 2023

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.

These are the Heinz numbers of the partitions counted by A367219.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        43: {14}        85: {3,7}
     5: {3}        47: {15}        89: {24}
     7: {4}        49: {4,4}       91: {4,6}
    11: {5}        53: {16}        95: {3,8}
    13: {6}        55: {3,5}       97: {25}
    17: {7}        59: {17}        99: {2,2,5}
    19: {8}        61: {18}       101: {26}
    23: {9}        63: {2,2,4}    103: {27}
    25: {3,3}      65: {3,6}      107: {28}
    27: {2,2,2}    67: {19}       109: {29}
    29: {10}       71: {20}       113: {30}
    31: {11}       73: {21}       115: {3,9}
    35: {3,4}      77: {4,5}      117: {2,2,6}
    37: {12}       79: {22}       119: {4,7}
    41: {13}       83: {23}       121: {5,5}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227*
A000700 counts self-conjugate partitions, ranks A088902.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124506 appears to count combination-free subsets, differences of A326083.
A229816 counts partitions whose length is not a part, ranks A367107.
A304792 counts subset-sums of partitions, strict A365925.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A046663, A088314, A106529, A116861, A236912, A364345, A364346, A364347, A364350, A365073, A365312.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p], {k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i}, {k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]=={}&]

cp's OEIS Frontend

A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

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A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

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A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

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A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

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