A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 0, 2, 0, 0, 2, 1, 1, 2, 3, 1, 1, 2, 0, 2, 0, 1, 4, 1, 0, 1, 3, 0, 1, 1, 2, 3, 2, 0, 2, 2, 0, 1, 1, 1, 4, 2, 1, 3, 2, 0, 2, 3, 0, 3, 1, 1, 3, 0, 0, 2, 0, 1, 0, 1, 1, 5, 0, 0, 2, 2, 2, 2, 2, 0, 2, 4, 0, 1, 1, 0, 4, 2, 1, 2, 2, 0, 4

Offset: 0

Author

Gus Wiseman, Dec 12 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.

Examples

			The prime indices of prime indices of 427 = 2*213 + 1 are {{1,1},{1,2,2}}, with four ways to choose (1,2), so a(213) = 4.
The prime indices of prime indices of 1469 = 2*734 + 1 are {{1,2},{1,2,3}}, with four choices (1,2), (1,3), (2,1), (2,3), so a(734) = 4.
		

Crossrefs

The "extended" version below includes alternating zeros at even positions.

Extended positions of zeros are A355529, binary A367907.

The extended version for binary indices is A367905.

Extended positions of nonzeros are A368100, binary A367906.

Extended positions of ones are A368101, binary A367908.

The extended version without distinctness is A355741, for multisets A355744.

A058891 counts set-systems, covering A003465, connected A323818.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

Cf. A007716, A355739, A355740, A355745, A367901, A367902, A367903.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[prix/@prix[2n+1]], UnsameQ@@#&]],{n,0,100}]