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Table T(n,k) by antidiagonals of exponent of largest power of k-th prime which divides n.
9

%I #36 Mar 27 2021 22:47:39

%S 0,0,1,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,

%T 0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0

%N Table T(n,k) by antidiagonals of exponent of largest power of k-th prime which divides n.

%F T(n, k) = log(A060176(n, k))/log(A000040(k)) = k-th digit from right of A054841(n).

%e a(12,1) = 2 since 4 = 2^2 = p_1^2 divides 12 but 8 = 2^3 does not.

%e a(12,2) = 1 since 3 = p_2 divides 12 but 9 = 3^2 does not.

%e See also examples in A249344, which is transpose of this array.

%e The top-left corner of the array:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...

%e ...

%t T[n_, k_] := IntegerExponent[n, Prime[k]];

%t Table[T[n-k+1, k], {n, 1, 15}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 18 2019 *)

%o (Scheme)

%o (define (A060175 n) (A249344bi (A004736 n) (A002260 n)))

%o (define (A249344bi row col) (let ((p (A000040 row))) (let loop ((n col) (i 0)) (cond ((not (zero? (modulo n p))) i) (else (loop (/ n p) (+ i 1)))))))

%o ;; _Antti Karttunen_, Oct 28 2014

%o (Python)

%o from sympy import prime

%o def a(n, k):

%o p=prime(n)

%o i=z=0

%o while p**i<=k:

%o if k%(p**i)==0: z=i

%o i+=1

%o return z

%o for n in range(1, 10): print([a(n - k + 1, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jun 24 2017

%o (PARI) a(n, k) = valuation(n, prime(k)); \\ _Michel Marcus_, Jun 24 2017

%Y Transpose: A249344.

%Y Column 1: A007814.

%Y Column 2: A007949.

%Y Column 3: A112765.

%Y Column 4: A214411.

%Y Cf. also A002260, A004736, A054841, A060176, A085604, A090622, A115627, A249421, A249422.

%K easy,nonn,tabl

%O 1,10

%A _Henry Bottomley_, Mar 14 2001

%E Erroneous example corrected and more terms computed by _Antti Karttunen_, Oct 28 2014