A379310 Number of nonsquarefree prime indices of n.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Dec 27 2024

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 39 are {2,6}, so a(39) = 0.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.
		

Crossrefs

Positions of first appearances are A000420.

Positions of zero are A302478, counted by A073576 (strict A087188).

No squarefree parts: A379307, counted by A114374 (strict A256012).

One squarefree part: A379316, counted by A379308 (strict A379309).

A000040 lists the primes, differences A001223.

A005117 lists the squarefree numbers, differences A076259.

A008966 is the characteristic function for the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A061398 counts squarefree numbers between primes, zeros A068360.

A377038 gives k-th differences of squarefree numbers.

Other counts of prime indices:

- A257991 odd, see A000041, A000070, A066207, A349158.

- A257992 even, see A000009, A038348, A066208, A379317.

- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.

- A330944 nonprime, see A000586, A000607, A076610, A330945.

- A379300 composite, see A023895, A034891, A036497, A302540, A379301.

- A379311 old prime, see A204389, A320629, A379312-A379315.

Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A087436, A112929, A120327, A378086.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],Not@*SquareFreeQ]],{n,100}]

Formula

Totally additive with a(prime(k)) = A107078(k) = 1 - A008966(k).