A394053 Irregular triangle read by rows: T(n,k) is the number of positive integers <= n with k odd divisors.

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, 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, 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

Offset

1,2

Comments

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

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

Links

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Example

Triangle begins:
  -----------------
   n\k  1  2  3  4
  -----------------
   1 |  1;
   2 |  2;
   3 |  2, 1;
   4 |  3, 1;
   5 |  3, 2;
   6 |  3, 3;
   7 |  3, 4;
   8 |  4, 4;
   9 |  4, 4, 1;
  10 |  4, 5, 1;
  11 |  4, 6, 1;
  12 |  4, 7, 1;
  13 |  4, 8, 1;
  14 |  4, 9, 1;
  15 |  4, 9, 1, 1;
  16 |  5, 9, 1, 1;
  ...

Mathematica

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 *)

Crossrefs

Row sums give A000027.

Column 1 gives A070939, n >= 1.

Analogous to A394052 and A394054.

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

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

Sequence in context: A167413 A373198 A340985 * A259176 A237591 A359979

Adjacent sequences: A394050 A394051 A394052 * A394054 A394055 A394056

Keyword

nonn,tabf,look

Author

Omar E. Pol, Mar 12 2026

Status

approved