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.

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

Author

Amiram Eldar, Aug 19 2024

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			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
		

Links

• 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.

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

Formula

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).