A307719 - OEIS

%I #28 Jan 12 2021 21:37:06

%S 0,0,0,1,1,1,2,1,3,2,4,2,7,2,8,4,8,4,15,4,16,7,15,7,26,7,23,11,26,10,

%T 43,9,35,16,38,16,54,14,49,23,54,18,79,18,66,31,64,25,100,25,89,36,85,

%U 31,127,35,104,46,104,39,167,36,125,58,129,52,185,45

%N Number of partitions of n into 3 mutually coprime parts.

%C The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - _Gus Wiseman_, Oct 16 2020

%H Fausto A. C. Cariboni, <a href="/A307719/b307719.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from Robert Israel)

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

%F a(n > 2) = A220377(n) + 1. - _Gus Wiseman_, Oct 15 2020

%e There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.

%e There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.

%e There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.

%e There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

%p N:= 200: # to get a(0)..a(N)

%p A:= Array(0..N):

%p for a from 1 to N/3 do

%p   for b from a to (N-a)/2 do

%p     if igcd(a,b) > 1 then next fi;

%p     ab:= a*b;

%p     for c from b to N-a-b do

%p        if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi

%p od od od:

%p convert(A,list); # _Robert Israel_, May 09 2019

%t Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]

%t Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* _Gus Wiseman_, Oct 15 2020 *)

%Y A023022 is the version for pairs.

%Y A220377 is the strict case, with ordered version A220377*6.

%Y A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.

%Y A337461 is the ordered version.

%Y A337563 is the case with no 1's.

%Y A337599 is the pairwise non-coprime instead of pairwise coprime version.

%Y A337601 only requires the distinct parts to be pairwise coprime.

%Y A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.

%Y A002865 counts partitions with no 1's, with strict case A025147.

%Y A007359 and A337485 count pairwise coprime partitions with no 1's.

%Y A200976 and A328673 count pairwise non-coprime partitions.

%Y Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

%K nonn,look

%O 0,7

%A _Wesley Ivan Hurt_, Apr 24 2019