%I #11 Dec 12 2021 15:58:53

%S 1,0,1,1,2,1,3,2,3,2,7,2,9,3,4,5,16,3,20,4,8,7,31,5,22,9,18,9,54,4,68,

%T 16,21,16,28,11,112,20,32,18,144,9,173,22,33,40,221,19,139,25,71,43,

%U 327,25,117,47,103,80,475,18,568,90,98,122,191,29,805,93,197,44

%N Number of partitions of n into distinct parts which are coprime to n and which are also pairwise relatively prime.

%C a(n) <= A036998(n); see A082415 for numbers m with a(m) = A036998(m).

%H Andrew Howroyd, <a href="/A083290/b083290.txt">Table of n, a(n) for n = 1..200</a>

%e a(7) = 3 since 7 = 3+4 = 2+5 = 1+6; 7 = 1+2+4 does not count (A036998(7)=4).

%t a[n_] := a[n] = If[n == 1, 1, Module[{ip}, ip = IntegerPartitions[n, All, Select[Range[n - 1], CoprimeQ[#, n] &]]; Length@Select[ip, Sort[#] == Union[#] && AllTrue[Subsets[#, {2}], CoprimeQ @@ # &] &]]];

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 80}] (* _Jean-François Alcover_, Dec 12 2021 *)

%o (PARI) a(n)={local(Cache=Map()); my(recurse(r,p,k)=my(hk=[r,p,k],z); if(!mapisdefined(Cache,hk,&z), z=if(k==0, r==0, self()(r,p,k-1) + if(gcd(p,k)==1, self()(r-k, p*k, min(r-k,k-1)))); mapput(Cache, hk, z)); z); recurse(n,n,n)} \\ _Andrew Howroyd_, Apr 20 2021

%Y Cf. A000010, A000586, A007360.

%K nonn

%O 1,5

%A _Reinhard Zumkeller_, Apr 23 2003

%E Terms a(31) and beyond from _Andrew Howroyd_, Apr 20 2021