A375668 The maximum exponent in the prime factorization of the 7-rough numbers (A007775).
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
If[# == 1, 0, Max[FactorInteger[#][[;; , 2]]]] & /@ Select[Range[300], CoprimeQ[#, 30] &]
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PARI
lista(nmax) = print1(0, ", "); for(n = 2, nmax, if(gcd(n, 30) == 1, print1(vecmax(factor(n)[,2]), ", ")));
Formula
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - 1/((1-1/2^k) * (1-1/3^k) * (1-1/5^k) * zeta(k))) = 1.05546104674564363968... .
In general, the asymptotic mean of the maximum exponent in the prime factorization of the p-rough numbers (numbers that are not divisible by any prime smaller than p) is 1 + Sum_{k>=2} (1 - 1/(zeta(k) * Product_{primes q < p} (1-1/q^k))).