A060176 Square array A(n,k) = the largest power of k-th prime which divides n, read by by falling antidiagonals.

1, 1, 2, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset

1,3

Comments

Product of terms on row n is n.

Links

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

Formula

A(n, k) = A000040(k)^A060175(n, k).

Example

The top left corner of the array:
n\k |  1   2   3   4   5   6   7   8
----+---------------------------------
1   |  1,  1,  1,  1,  1,  1,  1,  1,
2   |  2,  1,  1,  1,  1,  1,  1,  1,
3   |  1,  3,  1,  1,  1,  1,  1,  1,
4   |  4,  1,  1,  1,  1,  1,  1,  1,
5   |  1,  1,  5,  1,  1,  1,  1,  1,
6   |  2,  3,  1,  1,  1,  1,  1,  1,
7   |  1,  1,  1,  7,  1,  1,  1,  1,
8   |  8,  1,  1,  1,  1,  1,  1,  1,
9   |  1,  9,  1,  1,  1,  1,  1,  1,
10  |  2,  1,  5,  1,  1,  1,  1,  1,
11  |  1,  1,  1,  1, 11,  1,  1,  1,
12  |  4,  3,  1,  1,  1,  1,  1,  1,
13  |  1,  1,  1,  1,  1, 13,  1,  1,
14  |  2,  1,  1,  7,  1,  1,  1,  1,
15  |  1,  3,  5,  1,  1,  1,  1,  1,
16  | 16,  1,  1,  1,  1,  1,  1,  1,
17  |  1,  1,  1,  1,  1,  1, 17,  1,
18  |  2,  9,  1,  1,  1,  1,  1,  1,
19  |  1,  1,  1,  1,  1,  1,  1, 19,
etc.
a(12,1) = 4 since 4 = 2^2 = prime(1)^2 divides 12 but 8 = 2^3 does not.

Prog

(PARI)
up_to = 105;
A060176sq(n, k) = (prime(k)^valuation(n, prime(k)));
A060176list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A060176sq(col, (a-(col-1))))); (v); };
v060176 = A060176list(up_to);
A060176(n) = v060176[n]; \\ Antti Karttunen, Jan 16 2025

Crossrefs

Columns include A006519, A038500.

Sequence in context: A361754 A298848 A193518 * A305297 A218220 A247487

Adjacent sequences: A060173 A060174 A060175 * A060177 A060178 A060179

Keyword

easy,nonn,tabl

Author

Henry Bottomley, Mar 14 2001

Extensions

Edited by Antti Karttunen, Jan 16 2025

Status

approved