A375038 -id:A375038 - cp's OEIS frontend

A375037 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.

0, 0, 2, 1, 3, 1, 0, 4, 1, 5, 1, 0, 0, 6, 1, 7, 1, 0, 4, 8, 1, 0, 9, 1, 2, 5, 10, 1, 11, 1, 2, 0, 0, 6, 12, 1, 13, 1, 2, 7, 14, 1, 0, 0, 15, 1, 2, 0, 8, 16, 1, 17, 1, 2, 0, 6, 9, 18, 1, 19, 1, 2, 0, 0, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 0, 0, 8, 12, 24

Offset: 1

Omar E. Pol, Jul 28 2024

The nonzero terms in row n are the nonmiddle divisors of n.

The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins:
  0;
  0, 2;
  1, 3;
  1, 0, 4;
  1, 5;
  1, 0, 0, 6;
  1, 7;
  1, 0, 4, 8;
  1, 0, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 0, 0, 6, 12;
  ...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors are [3, 4], but here the middle divisors are replaced with zeros, so the 12th row of the triangle is [1, 2, 0, 0, 6, 12].

Row sums give A302433.
Nonzero terms give A375038.
Row lengths give A000005.
The number of zeros in row n is A067742(n).
The number of nonzero terms in row n is A067743(n).
Cf. A000005, A027750, A067742, A299761, A303297.

row[n_] := Divisors[n] /. {x_?(Sqrt[n/2] <= # < Sqrt[2*n] &) -> 0}; Table[row[n], {n, 1, 24}] // Flatten (* Amiram Eldar, Jul 29 2024 *)

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A375037 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.

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