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%I #11 Jan 16 2025 11:31:01
%S 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 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 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 Square array A(n,k) = the largest power of k-th prime which divides n, read by by falling antidiagonals.
%C Product of terms on row n is n.
%H Antti Karttunen, <a href="/A060176/b060176.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%F A(n, k) = A000040(k)^A060175(n, k).
%e The top left corner of the array:
%e n\k | 1 2 3 4 5 6 7 8
%e ----+---------------------------------
%e 1 | 1, 1, 1, 1, 1, 1, 1, 1,
%e 2 | 2, 1, 1, 1, 1, 1, 1, 1,
%e 3 | 1, 3, 1, 1, 1, 1, 1, 1,
%e 4 | 4, 1, 1, 1, 1, 1, 1, 1,
%e 5 | 1, 1, 5, 1, 1, 1, 1, 1,
%e 6 | 2, 3, 1, 1, 1, 1, 1, 1,
%e 7 | 1, 1, 1, 7, 1, 1, 1, 1,
%e 8 | 8, 1, 1, 1, 1, 1, 1, 1,
%e 9 | 1, 9, 1, 1, 1, 1, 1, 1,
%e 10 | 2, 1, 5, 1, 1, 1, 1, 1,
%e 11 | 1, 1, 1, 1, 11, 1, 1, 1,
%e 12 | 4, 3, 1, 1, 1, 1, 1, 1,
%e 13 | 1, 1, 1, 1, 1, 13, 1, 1,
%e 14 | 2, 1, 1, 7, 1, 1, 1, 1,
%e 15 | 1, 3, 5, 1, 1, 1, 1, 1,
%e 16 | 16, 1, 1, 1, 1, 1, 1, 1,
%e 17 | 1, 1, 1, 1, 1, 1, 17, 1,
%e 18 | 2, 9, 1, 1, 1, 1, 1, 1,
%e 19 | 1, 1, 1, 1, 1, 1, 1, 19,
%e etc.
%e a(12,1) = 4 since 4 = 2^2 = prime(1)^2 divides 12 but 8 = 2^3 does not.
%o (PARI)
%o up_to = 105;
%o A060176sq(n,k) = (prime(k)^valuation(n,prime(k)));
%o 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 v060176 = A060176list(up_to);
%o A060176(n) = v060176[n]; \\ _Antti Karttunen_, Jan 16 2025
%Y Columns include A006519, A038500.
%K easy,nonn,tabl
%O 1,3
%A _Henry Bottomley_, Mar 14 2001
%E Edited by _Antti Karttunen_, Jan 16 2025