A375537 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,10

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.

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.

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).

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

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

Sequence in context: A077614 A336396 A280379 * A333307 A180793 A352523

Adjacent sequences: A375534 A375535 A375536 * A375538 A375539 A375540

KEYWORD

nonn,easy,tabl

AUTHOR

Amiram Eldar, Aug 19 2024

STATUS

approved