cp's OEIS Frontend

A375538 Numerator of the asymptotic mean over the positive integers of the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor function.

%I A375538 #8 Aug 20 2024 09:14:42
%S A375538 1,13,51227,926908275845,548123689541583443758024333411,
%T A375538 629375533747930240763697631488051776709110194920714685268467462860005271344878614119
%N A375538 Numerator of the asymptotic mean over the positive integers of the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor function.
%C A375538 The numbers of digits of the terms are 1, 2, 5, 12, 30, 84, 215, 537, 1237, 2930, 6775, 15484, 35185, ... .
%H A375538 Amiram Eldar, <a href="/A375538/b375538.txt">Table of n, a(n) for n = 1..8</a>
%H A375538 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375538 Let f(n) = a(n)/A375539(n). Then:
%F A375538 f(n) = lim_{m->oo} (1/m) * Sum_{i=1..m} A375537(n, i).
%F A375538 f(n) = Sum_{k>=1} k * (d(k+1, prime(n)) - d(k, prime(n))), where d(k, p) = Product_{q prime <= p} (1 - 1/q^k).
%F A375538 Limit_{n->oo} f(n) = A033150.
%e A375538 Fractions begins: 1, 13/10, 51227/36540, 926908275845/636617813832, 548123689541583443758024333411/369693143251781030056182487680, ...
%e A375538 For n = 1, prime(1) = 2, the "2-smooth numbers" are the powers of 2 (A000079), and the sequence that gives the exponent of the largest power of 2 that divides n is A007814, whose asymptotic mean is 1.
%e A375538 For n = 2, prime(2) = 3, the 3-smooth numbers are in A003586, and the sequence that gives the maximum exponent in the prime factorization of the largest 3-smooth divisor of n is A244417, whose asymptotic mean is 13/10.
%t A375538 d[k_, n_] := Product[1 - 1/Prime[i]^k, {i, 1, n}]; f[n_] := Sum[k * (d[k+1, n] - d[k, n]), {k, 1, Infinity}]; Numerator[Array[f, 6]]
%Y A375538 Cf. A033150, A375537, A375539 (denominators).
%Y A375538 Cf. A375538 (numerators).
%Y A375538 Cf. A000079, A003586, A007814, A244417, A375536.
%K A375538 nonn,frac
%O A375538 1,2
%A A375538 _Amiram Eldar_, Aug 19 2024