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A384234 Irregular triangle read by rows: T(n,k) is the index of the k-th odd noncomposite divisor in the list of divisors of n, with n >=1, k >= 1.

%I A384234 #12 Jun 03 2025 01:00:09
%S A384234 1,1,1,2,1,1,2,1,3,1,2,1,1,2,1,3,1,2,1,3,1,2,1,3,1,2,3,1,1,2,1,3,1,2,
%T A384234 1,4,1,2,3,1,3,1,2,1,3,1,2,1,3,1,2,1,4,1,2,1,3,4,1,2,1,1,2,3,1,3,1,2,
%U A384234 3,1,3,1,2,1,3,1,2,3,1,4,1,2,1,3,5,1,2,1,4,1,2,3,1,3,1,2,1,3,1,2,1,3
%N A384234 Irregular triangle read by rows: T(n,k) is the index of the k-th odd noncomposite divisor in the list of divisors of n, with n >=1, k >= 1.
%C A384234 Row n lists the indices of the odd noncomposite divisors in the list of divisors of n.
%C A384234 Row n is [1] if and only if n is a power of 2 (A000079).
%e A384234 Triangle begins (n = 1..21):
%e A384234   1;
%e A384234   1;
%e A384234   1, 2;
%e A384234   1;
%e A384234   1, 2;
%e A384234   1, 3;
%e A384234   1, 2;
%e A384234   1;
%e A384234   1, 2;
%e A384234   1, 3;
%e A384234   1, 2;
%e A384234   1, 3;
%e A384234   1, 2;
%e A384234   1, 3;
%e A384234   1, 2, 3;
%e A384234   1;
%e A384234   1, 2;
%e A384234   1, 3;
%e A384234   1, 2;
%e A384234   1, 4;
%e A384234   1, 2, 3;
%e A384234   ...
%e A384234 For n = 30 the divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30] and the odd noncomposite divisors are [1, 3, 5] and the indices of them in the list of divisors are [1, 3, 4] respectively, so the 30th row of the triangle is [1, 3, 4].
%t A384234 row[n_] := Module[{m = n/2^IntegerExponent[n, 2]}, Join[{1}, If[m == 1, {}, Position[Divisors[n], #] & /@ FactorInteger[m][[;; , 1]] // Flatten]]]; Array[row, 50] // Flatten (* _Amiram Eldar_, May 29 2025 *)
%Y A384234 Companion of A383962.
%Y A384234 Column 1 gives A000012.
%Y A384234 Right border gives A384231.
%Y A384234 Cf. A006005 (odd noncomposite numbers).
%Y A384234 Cf. A000079, A027750, A384232, A384233.
%K A384234 nonn,tabf,easy
%O A384234 1,4
%A A384234 _Omar E. Pol_, May 29 2025