A379300 Number of prime indices of n that are composite.

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

Offset

1,49

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Totally additive with a(prime(k)) = A066247(k).

a(n) = A001222(n) - A379311(n). - Amiram Eldar, Jan 17 2026

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) = 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) = 3.
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], CompositeQ]], {n, 100}]

Prog

(PARI) A066247(n) = n > 1 && !isprime(n);
a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * A066247(primepi(f[i, 1]))); } \\ Amiram Eldar, Jan 17 2026

Crossrefs

Positions of first appearances are A000420.

Positions of zero are A302540, counted by A034891 (strict A036497).

Positions of one are A379301, counted by A379302 (strict A379303).

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.

A066247 is the characteristic function for the composite numbers.

A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.

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 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.

- A379311 old prime, see A379312-A379315.

Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

Sequence in context: A257402 A359966 A230263 * A139354 A124762 A258590

Adjacent sequences: A379297 A379298 A379299 * A379301 A379302 A379303

Keyword

nonn,easy

Author

Gus Wiseman, Dec 25 2024

Status

approved