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.

A325032 Product of products of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 4, 1, 1, 2, 1, 3, 4, 1, 4, 2, 1, 1, 3, 2, 5, 1, 3, 4, 2, 1, 2, 1, 2, 2, 6, 1, 4, 3, 2, 4, 6, 1, 1, 4, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 4, 5, 1, 1, 4, 3, 8, 4, 4, 2, 3, 1, 8, 2, 4, 1, 3, 2, 5, 2, 1, 6, 9, 1, 8, 4, 3
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2019

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.

Examples

			94 has prime indices {1,15} with prime indices {{},{2,3}} with product a(94) = 6.

Crossrefs

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A320325.
Cf. A324850, A324926, A324930, A325031, A325033, A325034, A325035.

Programs

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

Formula

Fully multiplicative with a(prime(n)) = A003963(n).

A324926 Numbers not divisible by any prime indices of their prime indices.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 16, 17, 22, 23, 25, 31, 32, 34, 41, 44, 47, 55, 59, 62, 64, 67, 73, 82, 83, 85, 88, 97, 103, 109, 115, 118, 121, 124, 125, 127, 128, 134, 137, 149, 157, 164, 166, 167, 176, 179, 187, 191, 194, 197, 205, 211, 218, 227, 233, 235, 236, 241, 242
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 55 are {3,5} with prime indices {{2},{3}}. Since 55 is not divisible by 2 or 3, it belongs to the sequence.

Examples

			The sequence of multisets of multisets whose MM-numbers (see A302242) belong to the sequence begins:
   1: {}
   2: {{}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  22: {{},{3}}
  23: {{2,2}}
  25: {{2},{2}}
  31: {{5}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  41: {{6}}
  44: {{},{},{3}}
  47: {{2,3}}
  55: {{2},{3}}
  59: {{7}}
  62: {{},{5}}
  64: {{},{},{},{},{},{}}

Links

Crossrefs

Cf. A001222, A003963, A112798, A120383, A302242, A304360, A324846, A324847, A324848, A324849, A324850, A324927, A324928, A324930.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@Table[!Divisible[#,i],{i,Union@@primeMS/@primeMS[#]}]&]

A324928 Matula-Goebel numbers of rooted trees of depth 3.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 23, 25, 26, 30, 34, 35, 37, 39, 40, 43, 45, 46, 50, 51, 52, 60, 61, 65, 67, 68, 69, 70, 73, 74, 75, 78, 80, 85, 86, 89, 90, 91, 92, 95, 100, 102, 103, 104, 105, 107, 111, 115, 117, 119, 120, 122, 125, 129, 130, 134, 135, 136, 138, 140
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

Numbers n such that A109082(n) = 3.

Examples

			The sequence of all rooted trees of depth 3 together with their Matula-Goebel numbers begins:
   5: (((o)))
  10: (o((o)))
  13: ((o(o)))
  15: ((o)((o)))
  17: (((oo)))
  20: (oo((o)))
  23: (((o)(o)))
  25: (((o))((o)))
  26: (o(o(o)))
  30: (o(o)((o)))
  34: (o((oo)))
  35: (((o))(oo))
  37: ((oo(o)))
  39: ((o)(o(o)))
  40: (ooo((o)))
  43: ((o(oo)))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  50: (o((o))((o)))
  51: ((o)((oo)))
  52: (oo(o(o)))
  60: (oo(o)((o)))

Crossrefs

Cf. A000081, A003963, A007097, A102378, A318400, A324927, A324930.

Programs

  • Mathematica
    Select[Range[100],Length[NestWhileList[Times@@PrimePi/@FactorInteger[#][[All,1]]&,#,#>1&]]-1==3&]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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