A375538 -id:A375538 - cp's OEIS frontend

A375539 Denominator 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, 10, 36540, 636617813832, 369693143251781030056182487680, 418823586043433867400108534336212749520449347490879717721482735332151276111078704000

Offset: 1

Amiram Eldar, Aug 19 2024

The numbers of digits of the terms are 1, 2, 5, 12, 30, 84, 215, 537, 1237, 2929, 6775, 15483, 35184, ... .

Cf. A375538 (numerators).

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}]; Denominator[Array[f, 6]]

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...

A375537 Square array A(n, k) (n, k >= 1) read by antidiagonals in ascending order: A(n, k) = Max_{i = 1..n} v_prime(i)(k), where v_p(k) is the p-adic valuation of k.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 1, 1, 2, 0, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 0, 3, 0, 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 0, 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 0, 2, 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 0, 2, 0, 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 0, 2, 0, 1

Offset: 1

Amiram Eldar, Aug 19 2024

For a given n, A(n, k) is the sequence that gives the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor of k.

			Array begins:
   n | n-th row
  ---+-----------------------------
   1 | 0, 1, 0, 2, 0, 1, 0, 3, 0, 1
   2 | 0, 1, 1, 2, 0, 1, 0, 3, 2, 1
   3 | 0, 1, 1, 2, 1, 1, 0, 3, 2, 1
   4 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   5 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   6 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   7 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   8 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   9 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
  10 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1

Amiram Eldar, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000720, A006530, A051903, A249344, A375538, A375539.

Rows k = 1..3: A007814, A244417, A375536.

A[n_, k_] := Max[IntegerExponent[k, Prime[Range[n]]]]; Table[A[n - k + 1, k], {n, 1, 14}, {k, 1 n}] // Flatten

A(n, k) = vecmax(apply(x -> valuation(k, x), primes(n)));

A(n, k) = Max_{i=1..n} A249344(i, k).
A(n, k) = A051903(k) for n >= A000720(A006530(k)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{i=1..m} A(n, i) = A375538(n)/A375539(n).

cp's OEIS Frontend

A375539 Denominator 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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A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

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A375537 Square array A(n, k) (n, k >= 1) read by antidiagonals in ascending order: A(n, k) = Max_{i = 1..n} v_prime(i)(k), where v_p(k) is the p-adic valuation of k.

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