A364906 Number of ways to write A056239(n) as a nonnegative linear combination of the multiset of prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Aug 22 2023

Keywords

• nonn

Comments

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.

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).

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

Examples

			The a(2) = 1 through a(10) = 2 ways:
  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
            1*1+1*1       3*1+0*2       0*1+1*1+2*1  1*2+1*2  4*1+0*3
            2*1+0*1                     0*1+2*1+1*1  2*2+0*2
                                        0*1+3*1+0*1
                                        1*1+0*1+2*1
                                        1*1+1*1+1*1
                                        1*1+2*1+0*1
                                        2*1+0*1+1*1
                                        2*1+1*1+0*1
                                        3*1+0*1+0*1
		

Crossrefs

The case with no zero coefficients is A000012.

Positions of 1's appear to be A319319.

A001222 counts prime indices, distinct A001221.

A112798 lists prime indices, sum A056239.

A364910 counts nonnegative linear combinations of strict partitions.

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[combs[Total[prix[n]],prix[n]]],{n,100}]