A276077 Number of distinct prime factors p of n such that p^(1+A000720(p)) is a divisor of n, where A000720(p) gives the index of prime p, 1 for 2, 2 for 3, 3 for 5, and so on.

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

Offset: 1

Author

Antti Karttunen, Aug 18 2016

Keywords

• nonn
• easy

Examples

			For n = 2 (= prime(1)), 2 is not divisible by 2^(1+1), thus a(2) = 0.
For n = 3 (= prime(3)), 3 is not divisible by 3^(2+1), thus a(3) = 0.
For n = 4 (= prime(1)^2), 4 is divisible by 2^(1+1), and there are no other prime factors apart from 2, thus a(4) = 1.
For n = 108 = 2^2 * 3^3, it is divisible both by 2^(1+1) and 3^(2+1), thus a(108) = 2.
For n = 625 = prime(3)^4, it is divisible by 5^(3+1), thus a(625) = 1.
		

Links

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

Crossrefs

Cf. A000720, A028234, A055396, A067029.

Cf. A276078 (positions of zeros), A276079 (nonzeros), also A276076.

Differs from A129251 for the first time at n=625, where a(625) = 1, while A129251(625) = 0.

Programs

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 *)

PARI

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i,1]) < f[i,2]);} \\ Amiram Eldar, Sep 30 2023

Python

from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
    f=factorint(n)
    return 1 if n==1 else n//(min(f)**f[min(f)])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n)*(isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n):
    if n==1: return 0
    val = a(a028234(n))
    if a067029(n) > a055396(n):
        val += 1
    return val
print([a(n) for n in range(1, 201)]) # Indranil Ghosh, Jun 21 2017

Scheme

(define (A276077 n) (if (= 1 n) 0 (+ (A276077 (A028234 n)) (if (> (A067029 n) (A055396 n)) 1 0))))

Formula

This formula uses Iverson bracket, which gives 1 as its value if the condition given inside [ ] is true and 0 otherwise:

a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) > A055396(n)].

Other identities. For all n >= 1:

a(A276076(n)) = 0.

From Amiram Eldar, Sep 30 2023: (Start)

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/prime(k)^(k+1) = 0.2886971166123417096098... . (End)