cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A382215 MM-numbers of multiset partitions into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 35, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2025

Keywords

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, sum A056239. 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}

Crossrefs

Twice-partitions of this type are counted by A279789.
For just constant blocks we have A302492, counted by A000688.
For sets of constant multisets we have A302496, counted by A050361.
For just common sums we have A326534, counted by A321455.
Factorizations of this type are counted by A381995.
For strict blocks and distinct sums we have A382201, counted by A381633.
Normal multiset partitions of this type are counted by A382204.
For strict instead of constant blocks we have A382304, counted by A382080.
For sets of constant multisets with distinct sums A382426, counted by A381635.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A000720, A000961, A001055, A302242, A302601, A368100, A381715, A381716, A381636, A381871.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#] && And@@SameQ@@@prix/@prix[#]&]
  • PARI
    is(k) = my(f=factor(k)[, 1]~, k, p, v=vector(#f, i, primepi(f[i]))); for(i=1, #v, k=isprimepower(v[i], &p); if(k||v[i]==1, v[i]=k*primepi(p), return(0))); #Set(v)<2; \\ Jinyuan Wang, Apr 02 2025

Formula

Equals A326534 /\ A302492.

A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 35, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599
Offset: 1

Views

Author

Gus Wiseman, Apr 25 2025

Keywords

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. 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. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The systems with these MM-numbers begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}

Crossrefs

Twice-partitions of this type are counted by A279789.
For just a common sum we have A326534.
For just constant blocks we have A355743.
Numbers without a factorization of this type are listed by A381871, counted by A381993.
The multiplicative version is A381995.
This is the odd case of A382215.
For strict instead of constant blocks we have A382304.
A001055 counts factorizations, strict A045778.
A023894 counts partitions into prime-powers.
A034699 gives maximal prime-power divisor.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
A355742 chooses a prime-power divisor of each prime index.
Cf. A000688, A000720, A001222, A006171, A038041, A279784, A302242, A302493, A321455, A326518, A381719.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@PrimePowerQ/@prix[#]&]

Formula

Equals A326534 /\ A355743.

A382426 MM-numbers of sets of constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 77, 82, 83, 85, 93, 95, 97, 102, 103, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159, 161, 165, 166
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2025

Keywords

Comments

Also products of prime numbers of prime power index with distinct sums 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, sum A056239. 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   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}}

Crossrefs

Twice-partitions of this type are counted by A279786.
For just constant blocks we have A302492.
For just distinct sums we have A326535.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A382201.
Normal multiset partitions of this type are counted by A382203.
For equal instead of distinct sums we have A382215.
An opposite version is A382304.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A000961, A302242, A302494, A321469, A326534, A355743, A356065, A381636, A381716.

Programs

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

Formula

Equals A302492 /\ A326535.
Showing 1-3 of 3 results.

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