A302505 -id:A302505 - cp's OEIS frontend

A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644

Offset: 1

Gus Wiseman, Dec 17 2018

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 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}}. This sequence lists all MM-numbers of 2-regular multiset multisystems (meaning all vertex-degrees are 2).

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   14: {{},{1,1}}
   18: {{},{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   28: {{},{},{1,1}}
   36: {{},{},{1},{1}}
   46: {{},{2,2}}
   50: {{},{2},{2}}
   56: {{},{},{},{1,1}}
   72: {{},{},{},{1},{1}}
   92: {{},{},{2,2}}
   97: {{3,3}}
  100: {{},{},{2},{2}}
  112: {{},{},{},{},{1,1}}
  121: {{3},{3}}
  144: {{},{},{},{},{1},{1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  184: {{},{},{},{2,2}}
  185: {{2},{1,1,2}}
  194: {{},{3,3}}
  195: {{1},{2},{1,2}}
  200: {{},{},{},{2},{2}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601

Offset: 1

Gus Wiseman, Dec 17 2018

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 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}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

			The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   97: {{3,3}}
  121: {{3},{3}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  185: {{2},{1,1,2}}
  195: {{1},{2},{1,2}}
  207: {{1},{1},{2,2}}
  225: {{1},{1},{2},{2}}
  227: {{4,4}}
  289: {{4},{4}}
  541: {{1,1,3,3}}
  661: {{5,5}}
  679: {{1,1},{3,3}}
  687: {{1},{1,3,3}}
  781: {{3},{1,1,3}}
  841: {{1,3},{1,3}}
  847: {{1,1},{3},{3}}
  873: {{1},{1},{3,3}}
  957: {{1},{3},{1,3}}
  961: {{5},{5}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322846 Squarefree numbers whose prime indices have no equivalent primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 74, 77, 78, 82, 83, 85, 87, 89, 91, 93, 95, 97, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Dec 28 2018

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.
In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.
Also MM-numbers of strict T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. 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}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).

			The sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  38: {{},{1,1,1}}
  39: {{1},{1,2}}

Cf. A000009, A005117, A056239, A059201, A112798, A302242, A302505, A316978, A316979, A316983, A319558, A319564, A319728, A322847.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Select[Range[100],And[SquareFreeQ[#],UnsameQ@@dual[primeMS/@primeMS[#]]]&]

A357458 First differences of A325033 = "Sum of sums of the multiset of prime indices of each prime index of n.".

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 30 2022

sign

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}}.

			We have A325033(5) - A325033(4) = 2 - 0, so a(4) = 2.

The partial sums are A325033, which has row-products A325032.
The version for standard compositions is A357187.
A000961 lists prime powers.
A003963 multiples prime indices.
A005117 lists squarefree numbers.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A109082, A275024, A302242, A302243, A302505, A324926, A325034, A357139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Differences[Table[Plus@@Join@@primeMS/@primeMS[n],{n,100}]]

a(n) = A325033(n + 1) - A325033(n).

A357188 Numbers with (WLOG adjacent) prime indices x <= y such that the greatest prime factor of x is greater than the least prime factor of y.

Original entry on oeis.org

35, 65, 70, 95, 105, 130, 140, 143, 145, 169, 175, 185, 190, 195, 209, 210, 215, 245, 247, 253, 260, 265, 280, 285, 286, 290, 305, 315, 319, 323, 325, 338, 350, 355, 370, 377, 380, 385, 390, 391, 395, 407, 418, 420, 429, 430, 435, 445, 455, 473, 475, 481, 490

Offset: 1

Gus Wiseman, Sep 30 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 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}}.

			The terms and corresponding multisets of multisets:
   35: {{2},{1,1}}
   65: {{2},{1,2}}
   70: {{},{2},{1,1}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  130: {{},{2},{1,2}}
  140: {{},{},{2},{1,1}}
  143: {{3},{1,2}}
  145: {{2},{1,3}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  185: {{2},{1,1,2}}

These are the positions of non-weakly increasing rows in A357139.
A000961 lists prime powers.
A003963 multiples prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A275024, A302242, A302243, A302505, A324926, A325032, A325034.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[primeMS[#],{_,x_,y_,_}/;Max@@primeMS[x]>Min@@primeMS[y]]&]
Select[Range[100],!LessEqual@@Join@@primeMS/@primeMS[#]&]

cp's OEIS Frontend

A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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A322846 Squarefree numbers whose prime indices have no equivalent primes.

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A357458 First differences of A325033 = "Sum of sums of the multiset of prime indices of each prime index of n.".

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A357188 Numbers with (WLOG adjacent) prime indices x <= y such that the greatest prime factor of x is greater than the least prime factor of y.

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