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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.

%I A375537 #11 Aug 20 2024 09:14:47
%S A375537 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,
%T A375537 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,
%U A375537 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
%N 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.
%C A375537 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.
%H A375537 Amiram Eldar, <a href="/A375537/b375537.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals, flattened)
%H A375537 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375537 A(n, k) = Max_{i=1..n} A249344(i, k).
%F A375537 A(n, k) = A051903(k) for n >= A000720(A006530(k)).
%F A375537 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{i=1..m} A(n, i) = A375538(n)/A375539(n).
%e A375537 Array begins:
%e A375537    n | n-th row
%e A375537   ---+-----------------------------
%e A375537    1 | 0, 1, 0, 2, 0, 1, 0, 3, 0, 1
%e A375537    2 | 0, 1, 1, 2, 0, 1, 0, 3, 2, 1
%e A375537    3 | 0, 1, 1, 2, 1, 1, 0, 3, 2, 1
%e A375537    4 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537    5 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537    6 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537    7 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537    8 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537    9 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%e A375537   10 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
%t A375537 A[n_, k_] := Max[IntegerExponent[k, Prime[Range[n]]]]; Table[A[n - k + 1, k], {n, 1, 14}, {k, 1 n}] // Flatten
%o A375537 (PARI) A(n, k) = vecmax(apply(x -> valuation(k, x), primes(n)));
%Y A375537 Cf. A000720, A006530, A051903, A249344, A375538, A375539.
%Y A375537 Rows k = 1..3: A007814, A244417, A375536.
%K A375537 nonn,easy,tabl
%O A375537 1,10
%A A375537 _Amiram Eldar_, Aug 19 2024