A375537 -id:A375537 - 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.

1, 13, 51227, 926908275845, 548123689541583443758024333411, 629375533747930240763697631488051776709110194920714685268467462860005271344878614119

Offset: 1

Amiram Eldar, Aug 19 2024

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

			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.

Cf. A033150, A375537, A375539 (denominators).
Cf. A375538 (numerators).
Cf. A000079, A003586, A007814, A244417, A375536.

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]]

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.

A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 2, 1, 3, 2, 0, 1, 0, 5, 1, 1, 1, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 6, 1, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 4, 1, 0, 2, 1, 1, 1, 3, 0, 2, 0, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 1, 0, 3, 1

Offset: 1

Amiram Eldar, Aug 19 2024

Cf. A007775, A051903, A244417 (3-smooth analog), A355582, A375537, A375538, A375539.

Cf. A007814, A007949, A112765.

a[n_] := Max[IntegerExponent[n, {2, 3, 5}]]; Array[a, 100]

a(n) = max(max(valuation(n, 2), valuation(n, 3)), valuation(n, 5));

a(n) = A051903(A355582(n)).
a(n) = max(A007814(n), A007949(n), A112765(n)).
a(n) = 0 if and only if n is a 7-rough number (A007775).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A375538(3)/A375539(3) = 51227/36540 = 1.401943076...

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.

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