A327406 -id:A327406 - cp's OEIS frontend

A327656 Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 5, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 1, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Gus Wiseman, Sep 21 2019

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. Numbers whose prime indices have a common divisor > 1 are listed in A318978, which is the union of this sequence without 1.

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

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

The union consists of {1} followed by A318978.
See link for additional cross-references.
Cf. A000005, A006530, A056239, A112798, A281116, A289509, A327406, A327657.

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

A327656(n) = vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))); \\ Antti Karttunen, Dec 06 2021

a(n) = n/A327405(n).

n belongs to A318978 iff a(n) = 1.

A327657 Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 3, 2, 1, 3, 2, 3, 3, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 3, 2, 2, 6, 1, 4, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 2, 3, 4, 2, 2, 5, 2, 2, 4, 3, 2, 4, 2, 2, 4, 4, 2, 3, 2, 3, 2, 2, 3, 4, 3, 2, 3, 2, 2, 5

Offset: 1

Gus Wiseman, Sep 21 2019

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. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

			The divisors of 90 that are 1 or whose prime indices have a common divisor > 1 are {1, 3, 5, 9}, so a(90) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

See link for additional cross-references.

Cf. A000005, A056239, A112798, A281116, A289509, A318979, A327406, A327656.

Table[Length[Select[Divisors[n],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]],{n,100}]

A327657(n) = sumdiv(n, d, (1==d)||(gcd(apply(x->primepi(x), factor(d)[, 1]))>1)); \\ Antti Karttunen, Dec 05 2021

a(n) = A000005(n) - A318979(n). - Antti Karttunen, Dec 05 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 05 2021

A327658 Number of factorizations of n that are empty or whose factors have a common divisor > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 21 2019

nonn

First differs from A319786 at a(900) = 11, A319786(900) = 12.

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 whose prime indices have a common divisor > 1 are listed in A318978.

			The a(120) = 7 factorizations:
  (120)
  (2*60)
  (4*30)
  (6*20)
  (10*12)
  (2*2*30)
  (2*6*10)

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A056239, A112798, A281116, A289509, A327406.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||GCD@@#!=1&]],{n,100}]

A327659 Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions.

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, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 7, 2, 1, 1

Offset: 1

Gus Wiseman, Sep 21 2019

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. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A056239, A112798, A281116, A289509, A318721, A327406.

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[1000],GCD@@PrimePi/@First/@FactorInteger[#]!=1&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

a(n) = A001055(A318978(n - 1)).

cp's OEIS Frontend

A327656 Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

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A327657 Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.

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Extensions

A327658 Number of factorizations of n that are empty or whose factors have a common divisor > 1.

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Keywords

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Examples

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Programs

Mathematica

A327659 Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions.

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Crossrefs

Programs

Mathematica

Formula