A276935 Number of distinct prime factors prime(k) of n such that prime(k)^k, but not prime(k)^(k+1) is a divisor of n.

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

Offset

1,18

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) = A055396(n)], where [] is Iverson bracket, giving 1 as its result when the stated equivalence is true and 0 otherwise.

From Amiram Eldar, Sep 30 2023: (Start)

Additive with a(p^e) = 1 if e = primepi(p), and 0 otherwise.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (1/prime(k)^k - 1/prime(k)^(k+1)) = 0.33083690651252383414... . (End)

Example

For n = 12 = 2*2*3 = prime(1)^2 * prime(2)^1, neither of the prime factors satisfies the condition, thus a(12) = 0.
For n = 18 = 2*3*3 = prime(1)^1 * prime(2)^2, both prime factors satisfy the condition, thus a(18) = 1+1 = 2.
For n = 750 = 2*3*5*5*5 = prime(1)^1 * prime(2)^1 * prime(3)^3, only the prime factors 2 and 5 satisfy the condition, thus a(750) = 1+1 = 2.

Mathematica

f[p_, e_] := If[PrimePi[p] == e, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 30 2023 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(definec (A276935 n) (if (= 1 n) 0 (+ (A276935 (A028234 n)) (if (= (A067029 n) (A055396 n)) 1 0))))
(PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i, 1]) == f[i, 2]); } \\ Amiram Eldar, Sep 30 2023

Crossrefs

Cf. A276077, A276936.

Sequence in context: A370813 A129251 A276077 * A235127 A358345 A258059

Adjacent sequences: A276932 A276933 A276934 * A276936 A276937 A276938

Keyword

nonn,easy

Author

Antti Karttunen, Sep 24 2016

Status

approved