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A384223 Irregular triangle read by rows: T(n,k) is the sum of the k-th odd divisor and the next even divisors that are less than the next odd divisor of n, with n >= 1, k >= 1.

%I A384223 #20 Jun 16 2025 17:30:34
%S A384223 1,3,1,3,7,1,5,3,9,1,7,15,1,3,9,3,15,1,11,3,25,1,13,3,21,1,3,5,15,31,
%T A384223 1,17,3,9,27,1,19,7,35,1,3,7,21,3,33,1,23,3,57,1,5,25,3,39,1,3,9,27,7,
%U A384223 49,1,29,3,3,21,45,1,31,63,1,3,11,33,3,51,1,5,7,35,3,13,75,1,37,3,57,1,3,13,39,7,83
%N A384223 Irregular triangle read by rows: T(n,k) is the sum of the k-th odd divisor and the next even divisors that are less than the next odd divisor of n, with n >= 1, k >= 1.
%C A384223 If n is odd the row n lists the divisors of n.
%C A384223 If n is a power of 2 then row n is 2*n - 1.
%e A384223 Triangle begins:
%e A384223    1;
%e A384223    3;
%e A384223    1,  3;
%e A384223    7;
%e A384223    1,  5;
%e A384223    3,  9;
%e A384223    1,  7;
%e A384223   15;
%e A384223    1,  3,  9;
%e A384223    3, 15;
%e A384223    1, 11;
%e A384223    3, 25;
%e A384223    1, 13;
%e A384223    3, 21;
%e A384223    1,  3,  5,  15;
%e A384223   31;
%e A384223   ...
%e A384223 For n = 30 the list of divisors of 30 is [1, 2, 3, 5, 6, 10, 15, 30]. There are four sublists of divisors whose first term is odd. They are [1, 2], [3], [5, 6, 10], [15, 30]. The sum of the divisors in the sublists are respectively [3, 3, 21, 45], the same as the 30th row of the triangle.
%o A384223 (PARI) row(n) = {my(d = divisors(n), res = vector(#d / (valuation(n, 2)+1)), t = 1, s = 1); for(i = 2, #d, if(bitand(d[i], 1), res[t] = s; t++; s = d[i], s += d[i])); res[#res] = s; res} \\ _David A. Corneth_, Jun 08 2025
%Y A384223 Row sums give A000203.
%Y A384223 Row lengths give A001227.
%Y A384223 Companion of A384224.
%Y A384223 Cf. A000079, A027750, A237270, A237271, A237593, A279387, A384149.
%K A384223 nonn,tabf,easy
%O A384223 1,2
%A A384223 _Omar E. Pol_, Jun 03 2025