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A060175
Square array A(n,k) = exponent of the largest power of k-th prime which divides n, read by falling antidiagonals.
9
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, 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, 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
OFFSET
1,10
FORMULA
A(n, k) = log(A060176(n, k))/log(A000040(k)) = k-th digit from right of A054841(n).
EXAMPLE
a(12,1) = 2 since 4 = 2^2 = p_1^2 divides 12 but 8 = 2^3 does not.
a(12,2) = 1 since 3 = p_2 divides 12 but 9 = 3^2 does not.
See also examples in A249344, which is transpose of this array.
The top-left corner of the array:
n\k | 1 2 3 4 5 6 7 8
----+------------------------
1 | 0, 0, 0, 0, 0, 0, 0, 0,
2 | 1, 0, 0, 0, 0, 0, 0, 0,
3 | 0, 1, 0, 0, 0, 0, 0, 0,
4 | 2, 0, 0, 0, 0, 0, 0, 0,
5 | 0, 0, 1, 0, 0, 0, 0, 0,
6 | 1, 1, 0, 0, 0, 0, 0, 0,
7 | 0, 0, 0, 1, 0, 0, 0, 0,
8 | 3, 0, 0, 0, 0, 0, 0, 0,
9 | 0, 2, 0, 0, 0, 0, 0, 0,
10 | 1, 0, 1, 0, 0, 0, 0, 0,
11 | 0, 0, 0, 0, 1, 0, 0, 0,
12 | 2, 1, 0, 0, 0, 0, 0, 0,
...
MATHEMATICA
T[n_, k_] := IntegerExponent[n, Prime[k]];
Table[T[n-k+1, k], {n, 1, 15}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 18 2019 *)
PROG
(Scheme)
(define (A060175 n) (A249344bi (A004736 n) (A002260 n)))
(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)))))))
;; Antti Karttunen, Oct 28 2014
(Python)
from sympy import prime
def a(n, k):
p=prime(n)
i=z=0
while p**i<=k:
if k%(p**i)==0: z=i
i+=1
return z
for n in range(1, 10): print([a(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 24 2017
(PARI) a(n, k) = valuation(n, prime(k)); \\ Michel Marcus, Jun 24 2017
CROSSREFS
Transpose: A249344.
Column 1: A007814.
Column 2: A007949.
Column 3: A112765.
Column 4: A214411.
Row sums: A001222.
Sequence in context: A028629 A047751 A028706 * A358480 A321537 A218876
KEYWORD
easy,nonn,tabl
AUTHOR
Henry Bottomley, Mar 14 2001
EXTENSIONS
Erroneous example corrected and more terms computed by Antti Karttunen, Oct 28 2014
Name clarified by Antti Karttunen, Jan 16 2025
STATUS
approved