A379306 Number of squarefree prime indices of n.

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 1, 1, 2, 4, 1, 3, 0, 3, 1, 2, 0, 4, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 1, 4, 0, 1, 2, 4, 1, 2, 1, 3, 3, 1, 1, 5, 0, 3, 2, 3, 0, 4, 2, 3, 1, 2, 1, 4, 0, 2, 2, 6, 2, 3, 1, 3, 1, 2, 0, 5, 1, 1, 3, 2, 1, 3, 1, 5, 4, 2, 1, 3, 2, 2, 2

Offset

1,4

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.

Links

Table of n, a(n) for n=1..87.

Formula

Totally additive with a(prime(k)) = A008966(k).

Example

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

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[prix[n], SquareFreeQ]], {n, 100}]

Crossrefs

Positions of first appearances are A000079.

Positions of zero are A379307, counted by A114374 (strict A256012).

Positions of one are 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:

- A087436 postpositive, see A038550.

- 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.

- A379310 nonsquarefree, see A302478.

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

Cf. A000720, A013928, A057627, A068361, A070321, A071403, A072284, A112925, A112929, A120327, A377430, A378086.

Sequence in context: A356006 A258120 A379311 * A147786 A275019 A337835

Adjacent sequences: A379303 A379304 A379305 * A379307 A379308 A379309

Keyword

nonn,new

Author

Gus Wiseman, Dec 25 2024

Status

approved