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.

1, 13, 51227, 926908275845, 548123689541583443758024333411, 629375533747930240763697631488051776709110194920714685268467462860005271344878614119

Offset

1,2

Comments

The numbers of digits of the terms are 1, 2, 5, 12, 30, 84, 215, 537, 1237, 2930, 6775, 15484, 35185, ... .

Links

Amiram Eldar, Table of n, a(n) for n = 1..8

Index entries for sequences computed from exponents in factorization of n.

Formula

Let f(n) = a(n)/A375539(n). Then:

f(n) = lim_{m->oo} (1/m) * Sum_{i=1..m} A375537(n, i).

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).

Limit_{n->oo} f(n) = A033150.

Example

Fractions begins: 1, 13/10, 51227/36540, 926908275845/636617813832, 548123689541583443758024333411/369693143251781030056182487680, ...
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.
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.

Mathematica

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]]

Crossrefs

Cf. A033150, A375537, A375539 (denominators).

Cf. A375538 (numerators).

Cf. A000079, A003586, A007814, A244417, A375536.

Sequence in context: A203691 A220981 A258670 * A048917 A081317 A203675

Adjacent sequences: A375535 A375536 A375537 * A375539 A375540 A375541

Keyword

nonn,frac

Author

Amiram Eldar, Aug 19 2024

Status

approved