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

Author

Henry Bottomley, Mar 14 2001

Keywords

• easy
• nonn
• tabl

Comments

Product of terms on row n is n.

Examples

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

Links

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

Crossrefs

Columns include A006519, A038500.

Programs

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

Formula

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

Extensions

Edited by Antti Karttunen, Jan 16 2025