%I #6 Aug 24 2023 09:30:19

%S 1,1,1,3,1,2,1,10,3,2,1,9,1,2,1,35,1,6,1,9,2,2,1,34,3,2,10,10,1,7,1,

%T 126,1,2,1,30,1,2,2,39,1,6,1,11,3,2,1,130,3,6,1,12,1,20,1,46,2,2,1,31,

%U 1,2,9,462,2,7,1,13,1,6,1,120,1,2,4,14,1,7,1

%N Number of ways to write A056239(n) as a nonnegative linear combination of the multiset of prime indices of n.

%C 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.

%C A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

%C Conjecture: Positions of 1's are numbers whose distinct divisors all have different GCDs of prime indices, listed by A319319, counted by A319318.

%e The a(2) = 1 through a(10) = 2 ways:

%e 1*1 1*2 0*1+2*1 1*3 1*1+1*2 1*4 0*1+0*1+3*1 0*2+2*2 1*1+1*3

%e 1*1+1*1 3*1+0*2 0*1+1*1+2*1 1*2+1*2 4*1+0*3

%e 2*1+0*1 0*1+2*1+1*1 2*2+0*2

%e 0*1+3*1+0*1

%e 1*1+0*1+2*1

%e 1*1+1*1+1*1

%e 1*1+2*1+0*1

%e 2*1+0*1+1*1

%e 2*1+1*1+0*1

%e 3*1+0*1+0*1

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];

%t Table[Length[combs[Total[prix[n]],prix[n]]],{n,100}]

%Y The case with no zero coefficients is A000012.

%Y Positions of 1's appear to be A319319.

%Y A001222 counts prime indices, distinct A001221.

%Y A112798 lists prime indices, sum A056239.

%Y A364910 counts nonnegative linear combinations of strict partitions.

%Y Cf. A116861, A275972, A320340, A364350, A364839, A364912, A364914, A364916, A365002, A365003, A365004.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 22 2023