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A274532 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the absolute difference table of the divisors of n.

%I A274532 #17 Apr 01 2017 19:43:24
%S A274532 1,1,3,1,5,1,3,7,1,9,1,3,4,13,1,13,1,3,7,15,1,5,19,1,3,10,17,1,21,1,3,
%T A274532 4,5,11,28,1,25,1,3,16,25,1,5,7,41,1,3,7,15,31,1,33,1,3,4,13,14,47,1,
%U A274532 37,1,3,7,7,25,39,1,5,13,53,1,3,28,41,1,45,1,3,4,5,11,12,22,61,1,9,61,1,3,34,49,1,5,19,65
%N A274532 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the absolute difference table of the divisors of n.
%C A274532 If n is prime then row n contains only two terms: 1 and 2*n-1.
%C A274532 Row 2^k gives the first k+1 positive terms of A000225, k >= 0.
%C A274532 Note that this sequence is not the absolute values of A273262.
%C A274532 First differs from A273262 at a(41).
%e A274532 Triangle begins:
%e A274532 1;
%e A274532 1, 3;
%e A274532 1, 5;
%e A274532 1, 3, 7;
%e A274532 1, 9;
%e A274532 1, 3, 4, 13;
%e A274532 1, 13;
%e A274532 1, 3, 7, 15;
%e A274532 1, 5, 19;
%e A274532 1, 3, 10, 17;
%e A274532 1, 21;
%e A274532 1, 3, 4, 5, 11, 28;
%e A274532 1, 25;
%e A274532 1, 3, 16, 25;
%e A274532 1, 5, 7, 41;
%e A274532 1, 3, 7, 15, 31;
%e A274532 1, 33;
%e A274532 1, 3, 4, 13, 14, 47;
%e A274532 1, 37;
%e A274532 1, 3, 7, 7, 25, 39;
%e A274532 1, 5, 13, 53;
%e A274532 1, 3, 28, 41;
%e A274532 1, 45;
%e A274532 1, 3, 4, 5, 11, 12, 22, 61;
%e A274532 1, 9, 61;
%e A274532 1, 3, 34, 49;
%e A274532 1, 5, 19, 65;
%e A274532 ...
%e A274532 For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the absolute difference triangle of the divisors is
%e A274532 1, 2, 3, 6, 9, 18;
%e A274532 1, 1, 3, 3, 9;
%e A274532 0, 2, 0, 6;
%e A274532 2, 2, 6;
%e A274532 0, 4;
%e A274532 4;
%e A274532 The antidiagonal sums give [1, 3, 4, 13, 14, 47] which is also the 18th row of the irregular triangle.
%t A274532 Table[Map[Total, Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}], {1}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 27}] // Flatten (* _Michael De Vlieger_, Jun 27 2016 *)
%Y A274532 Row lengths give A000005. Column 1 is A000012. Row sums give A187215.
%Y A274532 Cf. A000225, A187203, A272121, A273132, A273135, A273262, A274531.
%K A274532 nonn,tabf
%O A274532 1,3
%A A274532 _Omar E. Pol_, Jun 27 2016