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.

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A327391 Number of divisors of n that are 1, prime, or whose prime indices are pairwise coprime.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 3, 8, 2, 6, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 4, 4, 8, 3, 4, 2, 12, 2, 4, 3, 7, 3, 8, 2, 6, 4, 8, 2, 8, 2, 4, 4, 6, 4, 6, 2, 10, 2, 4, 2, 9, 4, 4
Offset: 1

Views

Author

Gus Wiseman, Sep 20 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. Numbers that are prime or whose prime indices are pairwise coprime are listed in A302569.

Examples

			The divisors of 84 that are 1, prime, or whose prime indices are pairwise coprime are {1, 2, 3, 4, 6, 7, 12, 14, 28}, so a(84) = 9.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A006530, A033273, A056239, A112798, A302569, A304711, A327513.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Divisors[n],#==1||PrimeQ[#]||CoprimeQ@@primeMS[#]&]],{n,100}]

A327404 Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 32, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5
Offset: 1

Views

Author

Gus Wiseman, Sep 23 2019

Keywords

Comments

First differs from A327395 at a(195) = 65, A327395(195) = 195.
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

			The divisors of 90 that are 2 or whose prime indices have a common divisor > 1 are {1, 2, 3, 5, 9}, so a(90) = 90/9 = 10.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A006530, A056239, A112798, A289509, A302569, A327407, A327656.

Programs

  • Mathematica
    Table[n/Max[Select[Divisors[n],#==2||GCD@@PrimePi/@First/@FactorInteger[#]!=1&]],{n,100}]

A327518 Number of factorizations of A302696(n), the n-th number that is 1, 2, or a nonprime number with pairwise coprime prime indices, into factors > 1 satisfying the same conditions.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 2, 7, 1, 1, 1, 1, 4, 2, 1, 7, 1, 2, 1, 4, 1, 5, 1, 11, 2, 2, 1, 2, 1, 2, 1, 7, 1, 1, 1, 4, 2, 1, 1, 1, 12, 2, 4, 1, 2, 7, 2, 1, 1, 10, 1, 1, 2, 15, 5, 1, 4, 2, 5, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 12, 1, 2, 1, 1, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Sep 19 2019

Keywords

Examples

			The a(59) = 10 factorizations of 120 using the allowed factors, together with the corresponding multiset partitions of {1,1,1,2,3}:
  (2*2*2*15)  {{1},{1},{1},{2,3}}
  (2*2*30)    {{1},{1},{1,2,3}}
  (2*4*15)    {{1},{1,1},{2,3}}
  (2*6*10)    {{1},{1,2},{1,3}}
  (2*60)      {{1},{1,1,2,3}}
  (4*30)      {{1,1},{1,2,3}}
  (6*20)      {{1,2},{1,1,3}}
  (8*15)      {{1,1,1},{2,3}}
  (10*12)     {{1,3},{1,1,2}}
  (120)       {{1,1,1,2,3}}

Links

Crossrefs

See link for additional cross-references.
Cf. A001055, A056239, A112798, A302569, A302696, A304711, A305079, A327392.

Programs

  • Mathematica
    nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
y=Select[Range[nn],#==1||CoprimeQ@@primeMS[#]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

A327519 Number of factorizations of A305078(n - 1), the n-th connected number, into connected numbers > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 7, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 7, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Sep 21 2019

Keywords

Comments

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.

Examples

			The a(190) = 8 factorizations of 585 together with the corresponding multiset partitions of {2,2,3,6}:
  (3*3*5*13)  {{2},{2},{3},{6}}
  (3*3*65)    {{2},{2},{3,6}}
  (3*5*39)    {{2},{3},{2,6}}
  (3*195)     {{2},{2,3,6}}
  (5*9*13)    {{3},{2,2},{6}}
  (5*117)     {{3},{2,2,6}}
  (9*65)      {{2,2},{3,6}}
  (585)       {{2,2,3,6}}

Links

Crossrefs

See link for additional cross-references.
Cf. A286518, A302569, A304714, A304716, A305078, A305079, A327076.

Programs

  • Mathematica
    nn=100;
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
y=Select[Range[nn],Length[zsm[primeMS[#]]]==1&];
Table[Length[facsusing[y,n]],{n,y}]

A327905 FDH numbers of pairwise coprime sets.

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 38, 40, 42, 44, 46, 48, 50, 52, 55, 56, 57, 58, 62, 63, 66, 68, 70, 74, 75, 76, 77, 80, 82, 84, 86, 88, 91, 93, 94, 95, 96, 98, 99, 100, 104, 106, 110, 112, 114, 116, 118, 122, 123, 125, 126, 132
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict partition or finite set {y_1,...,y_k} is f(y_1)*...*f(y_k).
We use the Mathematica function CoprimeQ, meaning a singleton is not coprime unless it is {1}.

Examples

			The sequence of terms together with their corresponding coprime sets begins:
   2: {1}
   6: {1,2}
   8: {1,3}
  10: {1,4}
  12: {2,3}
  14: {1,5}
  18: {1,6}
  20: {3,4}
  21: {2,5}
  22: {1,7}
  24: {1,2,3}
  26: {1,8}
  28: {3,5}
  32: {1,9}
  33: {2,7}
  34: {1,10}
  35: {4,5}
  38: {1,11}
  40: {1,3,4}
  42: {1,2,5}

Links

  • Wolfram Language Documentation, CoprimeQ

Crossrefs

Heinz numbers of pairwise coprime partitions are A302696 (all), A302797 (strict), A302569 (with singletons), and A302798 (strict with singletons).
FDH numbers of relatively prime sets are A319827.
Cf. A050376, A056239, A064547, A213925, A259936, A299755, A299757, A304711, A319826, A326675.

Programs

  • Mathematica
    FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],CoprimeQ@@(FDfactor[#]/.FDrules)&]
Previous Showing 71-75 of 75 results.

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