A060175 Square array A(n,k) = exponent of the largest power of k-th prime which divides n, read by falling antidiagonals.

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

Author

Henry Bottomley, Mar 14 2001

Keywords

• easy
• nonn
• tabl

Examples

			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,
...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals

Crossrefs

Transpose: A249344.

Column 1: A007814.

Column 2: A007949.

Column 3: A112765.

Column 4: A214411.

Row sums: A001222.

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

Programs

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 *)

PARI

a(n, k) = valuation(n, prime(k)); \\ Michel Marcus, Jun 24 2017

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

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

Formula

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

Extensions

Erroneous example corrected and more terms computed by Antti Karttunen, Oct 28 2014

Name clarified by Antti Karttunen, Jan 16 2025