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

1,10

Links

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

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.

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

Sequence in context: A028629 A047751 A028706 * A358480 A321537 A218876

Adjacent sequences: A060172 A060173 A060174 * A060176 A060177 A060178

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