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.

A327535 Maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 7, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 19, 58, 59, 60, 61, 62, 7, 64, 13, 66, 67, 68, 69
Offset: 1

Views

Author

Gus Wiseman, Sep 17 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 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

Examples

			The divisors of 63 that are 1, prime, or whose prime indices are relatively prime are {1, 3, 7}, so a(63) = 7.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A000961, A006530, A056239, A112798, A281116, A289509, A327407.

Programs

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

Formula

If n is in A327534, then a(n) = n; otherwise a(n) = A006530(n).

A327538 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2019

Keywords

Comments

The first index m such that a(m) > 1 but m is not in A322336 is m = 2335.
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 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

Examples

			We have 441 -> 63 -> 9 -> 3 -> 1, so a(441) = 4.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A001222, A006530, A056239, A112798, A289509, A327407.

Programs

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

Formula

a(1) = 0; if n is prime or has relatively prime prime indices, then a(n) = 1; otherwise a(n) = Omega(n) = A001222(n).

A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 7, 1, 4, 1, 4, 3, 2, 1, 12, 3, 2, 4, 1, 4, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 2, 1, 9, 2, 2, 7, 1, 9, 4, 2, 2, 2, 19, 1
Offset: 1

Views

Author

Gus Wiseman, Sep 17 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 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

Examples

			The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
  (2*2*3*7)  {{1},{1},{2},{4}}
  (2*3*14)   {{1},{2},{1,4}}
  (2*6*7)    {{1},{1,2},{4}}
  (2*42)     {{1},{1,2,4}}
  (3*4*7)    {{2},{1,1},{4}}
  (3*28)     {{2},{1,1,4}}
  (6*14)     {{1,2},{1,4}}
  (7*12)     {{4},{1,1,2}}
  (84)       {{1,1,2,4}}

Links

Crossrefs

See link for additional cross-references.
Cf. A006530, A056239, A112798, A281116, A289509, A327407.

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||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y],n]],{n,y}]
Showing 1-3 of 3 results.

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