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

1,108

Links

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

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)

Example

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.

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

Sequence in context: A219488 A370813 A129251 * A276935 A235127 A358345

Adjacent sequences: A276074 A276075 A276076 * A276078 A276079 A276080

Keyword

nonn,easy

Author

Antti Karttunen, Aug 18 2016

Status

approved