A394053 - OEIS

%I #22 Mar 31 2026 19:09:42

%S 1,2,2,1,3,1,3,2,3,3,3,4,4,4,4,4,1,4,5,1,4,6,1,4,7,1,4,8,1,4,9,1,4,9,

%T 1,1,5,9,1,1,5,10,1,1,5,10,2,1,5,11,2,1,5,12,2,1,5,12,2,2,5,13,2,2,5,

%U 14,2,2,5,15,2,2,5,15,3,2,5,16,3,2,5,16,3,3,5,17,3,3,5,18,3,3,5,18,3,4

%N Irregular triangle read by rows: T(n,k) is the number of positive integers <= n with k odd divisors.

%C T(n,k) is also the number of positive integers <= n with k partitions into consecutive parts.

%C T(n,k) is also the number of positive integers m <= n whose symmetric representation of sigma(m) has k subparts.

%H Sean A. Irvine, <a href="/A394053/b394053.txt">Table of n, a(n) for n = 1..10000</a>

%e Triangle begins:

%e   -----------------

%e    n\k  1  2  3  4

%e   -----------------

%e    1 |  1;

%e    2 |  2;

%e    3 |  2, 1;

%e    4 |  3, 1;

%e    5 |  3, 2;

%e    6 |  3, 3;

%e    7 |  3, 4;

%e    8 |  4, 4;

%e    9 |  4, 4, 1;

%e   10 |  4, 5, 1;

%e   11 |  4, 6, 1;

%e   12 |  4, 7, 1;

%e   13 |  4, 8, 1;

%e   14 |  4, 9, 1;

%e   15 |  4, 9, 1, 1;

%e   16 |  5, 9, 1, 1;

%e   ...

%t T[nmax_] := Module[{t = Table[DivisorSigma[0, n/2^IntegerExponent[n, 2]], {n, 1, nmax}]}, Table[Tally[t[[1 ;; k]]][[;; , 2]], {k, 1, nmax}] // Flatten]; T[30] (* _Amiram Eldar_, Mar 12 2026 *)

%Y Row sums give A000027.

%Y Column 1 gives A070939, n >= 1.

%Y Analogous to A394052 and A394054.

%Y Cf. A053624 (row numbers where the triangle widens).

%Y Cf. A001227, A038547, A237593, A279387, A299765.

%K nonn,tabf,look

%O 1,2

%A _Omar E. Pol_, Mar 12 2026