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

%I A060176 #15 Jul 21 2025 08:56:19
%S A060176 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,
%T A060176 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,
%U A060176 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
%N A060176 Square array A(n,k) = the largest power of k-th prime which divides n, read by falling antidiagonals.
%C A060176 Product of terms on row n is n.
%H A060176 Antti Karttunen, <a href="/A060176/b060176.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%F A060176 A(n, k) = A000040(k)^A060175(n, k).
%e A060176 The top left corner of the array:
%e A060176   n\k |  1   2   3   4   5   6   7   8
%e A060176   ----+---------------------------------
%e A060176    1  |  1,  1,  1,  1,  1,  1,  1,  1,
%e A060176    2  |  2,  1,  1,  1,  1,  1,  1,  1,
%e A060176    3  |  1,  3,  1,  1,  1,  1,  1,  1,
%e A060176    4  |  4,  1,  1,  1,  1,  1,  1,  1,
%e A060176    5  |  1,  1,  5,  1,  1,  1,  1,  1,
%e A060176    6  |  2,  3,  1,  1,  1,  1,  1,  1,
%e A060176    7  |  1,  1,  1,  7,  1,  1,  1,  1,
%e A060176    8  |  8,  1,  1,  1,  1,  1,  1,  1,
%e A060176    9  |  1,  9,  1,  1,  1,  1,  1,  1,
%e A060176   10  |  2,  1,  5,  1,  1,  1,  1,  1,
%e A060176   11  |  1,  1,  1,  1, 11,  1,  1,  1,
%e A060176   12  |  4,  3,  1,  1,  1,  1,  1,  1,
%e A060176   13  |  1,  1,  1,  1,  1, 13,  1,  1,
%e A060176   14  |  2,  1,  1,  7,  1,  1,  1,  1,
%e A060176   15  |  1,  3,  5,  1,  1,  1,  1,  1,
%e A060176   16  | 16,  1,  1,  1,  1,  1,  1,  1,
%e A060176   17  |  1,  1,  1,  1,  1,  1, 17,  1,
%e A060176   18  |  2,  9,  1,  1,  1,  1,  1,  1,
%e A060176   19  |  1,  1,  1,  1,  1,  1,  1, 19,
%e A060176   ...
%e A060176 a(12,1) = 4 since 4 = 2^2 = prime(1)^2 divides 12 but 8 = 2^3 does not.
%o A060176 (PARI)
%o A060176 up_to = 105;
%o A060176 A060176sq(n,k) = (prime(k)^valuation(n,prime(k)));
%o A060176 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); };
%o A060176 v060176 = A060176list(up_to);
%o A060176 A060176(n) = v060176[n]; \\ _Antti Karttunen_, Jan 16 2025
%Y A060176 Columns include A006519, A038500.
%K A060176 easy,nonn,tabl
%O A060176 1,3
%A A060176 _Henry Bottomley_, Mar 14 2001
%E A060176 Edited by _Antti Karttunen_, Jan 16 2025