A085945 -id:A085945 - cp's OEIS frontend

A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 5, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 2, 0, 7, 1, 1, 1, 3, 0, 1, 0, 4, 0, 1, 0, 2, 1, 1, 0, 7, 0, 1, 1, 2, 0, 1, 1, 4, 0, 1, 0, 5, 0, 1, 0, 11, 0, 2, 0, 2, 1, 2, 0, 6, 0, 1, 1, 2, 1, 1, 0, 7, 0, 1, 0, 3, 1, 1, 0

Offset: 1

Gus Wiseman, Jan 08 2019

nonn

Also the number of multiset partitions of the multiset of prime indices of n using multisets each of which is relatively prime.
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.
Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

			The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of 0's are A318978.

Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facsrp[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsrp[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],GCD@@primeMS[#]==1&]}]];
Table[Length[facsrp[n]],{n,100}]

A320439(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d<=m)&&(1==gcd(apply(x->primepi(x), factor(d)[, 1]))), s += A320439(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2021

A143446 Triangle read by rows: A000012 * A143424.

Original entry on oeis.org

1, 0, 2, -1, 2, 4, -1, 0, 4, 8, -2, 0, 4, 8, 16, -1, -2, 0, 8, 16, 32, -2, -2, 0, 8, 16, 32, 64, -2, -2, 0, 0, 16, 32, 64, 128, -2, -2, -4, 0, 16, 32, 64, 128, 256, -1, -4, -4, 0, 0, 32, 64, 128, 256, 512, -2, -4, -4, 0, 0, 32, 64, 128, 256, 512, 1024, -2, -2, -4, -8, 0, 0, 64, 128, 256, 512, 1024, 2048

Offset: 1

Gary W. Adamson, Aug 15 2008

Row sums = A085945: (1, 2, 5, 11, 26, 53, 116, ...).

			First few rows of the triangle:
   1;
   0,  2;
  -1,  2,  4;
  -1,  0,  4,  8;
  -2,  0,  4,  8, 16;
  -1, -2,  0,  8, 16, 32;
  -2, -2,  0,  8, 16, 32, 64;
  ...

Cf. A085945, A143424.

a(37) = -2 corrected and more terms from Georg Fischer, Jun 05 2023

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 1

Gus Wiseman, Jan 09 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Ana Rechtman, Décembre 2020, 4e défi (in French), Images des Mathématiques, CNRS, 2020.

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)

a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

a(n) = Product_{p prime <= n} floor(log_p(n)).

a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

cp's OEIS Frontend

A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

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A143446 Triangle read by rows: A000012 * A143424.

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A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

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Mathematica

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