A327656 -id:A327656 - cp's OEIS frontend

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2

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.

Note that A318978 includes also all odd primes and their powers, thus the only numbers for which a maximum such divisor is 1 are the powers of 2. Therefore A000079 gives the indices of zeros in this sequence. - Antti Karttunen, Dec 06 2021

			We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

First appearance of n is A080696(n).
See link for additional cross-references.
Cf. A000005, A000079 (positions of 0's), A056239, A112798, A281116, A289509, A302569, A318978.

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

A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))));
A327406(n) = { my(u = A327405(n), k=0); while(u!=n, k++; n = u; u = A327405(n)); (k); }; \\ Antti Karttunen, Dec 06 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

A327398 Maximum connected squarefree divisor of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 7, 71, 3, 73, 37

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

A squarefree number with prime factorization prime(m_1) * ... * prime(m_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.

			The connected squarefree divisors of 189 are {1, 3, 7, 21}, so a(189) = 21.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

The maximum connected divisor of n is A327076(n).
The maximum squarefree divisor of n is A007947(n).
Connected numbers are A305078.
Connected squarefree numbers are A328513.
Cf. A000005, A001221, A302242, A302494, A304714, A305079, A327656, A328514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Max[Select[Divisors[n],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]],{n,100}]

A327405 Quotient of n over the maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 32, 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, 64, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5

Offset: 1

Gus Wiseman, Sep 21 2019

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) = 90/9 = 10.

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, A302569.

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

A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n)))); \\ Antti Karttunen, Dec 06 2021

a(n) = n/A327656(n).

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

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

Gus Wiseman, Sep 23 2019

nonn

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.

			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.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A289509, A302569, A327407, A327656.

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

cp's OEIS Frontend

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

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A327398 Maximum connected squarefree divisor of n.

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A327405 Quotient of n over the 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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A327404 Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1.

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