A379311 Number of prime indices of n that are 1 or prime.

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

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)) = A080339(k).

Example

The prime indices of 39 are {2,6}, so a(39) = 1.
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) = 1.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Mathematica

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

Crossrefs

Positions of first appearances are A000079.

These "old" primes are listed by A008578.

Positions of zero are A320629, counted by A023895 (strict A204389).

Positions of one are A379312, counted by A379314 (strict A379315).

Positions of nonzero terms are A379313.

A000040 lists the prime numbers, differences A001223.

A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.

A055396 gives least prime index, greatest A061395.

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

A080339 is the characteristic function for the old prime numbers.

A376682 gives k-th differences of old prime numbers, see A030016, A075526, A173390, A376683, A376855.

Other counts of prime indices:

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

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

- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.

- A330944 nonprime, see A002095, A096258, A320628, A330945.

- A379306 squarefree, see A302478, A379308, A379309, A379316.

- A379310 nonsquarefree, see A114374, A256012, A379307.

Cf. A034891, A036234, A036497, A038550, A087436, A302540, A379300, A379301.

Sequence in context: A238735 A356006 A258120 * A379306 A147786 A275019

Adjacent sequences: A379308 A379309 A379310 * A379312 A379313 A379314

Keyword

nonn,new

Author

Gus Wiseman, Dec 27 2024

Status

approved