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A138045 Triangle read by rows: largest proper divisor of n as a table, ones excluded.

%I A138045 #10 Feb 16 2025 08:33:07
%S A138045 0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,
%T A138045 0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A138045 0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0
%N A138045 Triangle read by rows: largest proper divisor of n as a table, ones excluded.
%C A138045 The numbers in the triangle form lines that begin at T(A001248,A000040). The first line of numbers from the right, is T(A005843,A000027). The second line is T(A016945,A005408). The third line is T(A084967,A007310).
%H A138045 Antti Karttunen, <a href="/A138045/b138045.txt">Table of n, a(n) for n = 1..23220</a> (the first 215 rows of the triangle).
%H A138045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProperDivisor.html">Proper Divisor</a>.
%F A138045 T(n,k) = if k==A032742(n) and n(T(n,k))==n(A032742(n)) and k>1 then k else 0 (1<=k<=n), T(1,1)=0.
%e A138045 The first few terms of the table are:
%e A138045   0
%e A138045   0,0
%e A138045   0,0,0
%e A138045   0,2,0,0
%e A138045   0,0,0,0,0
%e A138045   0,0,3,0,0,0
%e A138045   0,0,0,0,0,0,0
%e A138045   0,0,0,4,0,0,0,0
%e A138045   0,0,3,0,0,0,0,0,0
%o A138045 (PARI)
%o A138045 up_to = 23220; \\ binomial(215+1,2)
%o A138045 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A138045 A138045tr(n, k) = if((k>1) && (A032742(n)==k), k, 0);
%o A138045 A138045list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A138045tr(n,k))); (v); };
%o A138045 v138045 = A138045list(up_to);
%o A138045 A138045(n) = v138045[n]; \\ _Antti Karttunen_, Dec 24 2018
%Y A138045 Cf. A001248, A016945, A084967, A007310, A127093, A032742, A007775.
%K A138045 nonn,tabl
%O A138045 1,8
%A A138045 _Mats Granvik_, Mar 02 2008