A379946 Irregular triangle read by rows: T(n, k) is the denominator of the harmonic mean of all positive divisors of n except the k-th of them.

1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 5, 11, 1, 1, 7, 11, 13, 7, 2, 5, 2, 4, 13, 8, 17, 1, 1, 4, 11, 2, 5, 13, 9, 1, 1, 5, 17, 11, 23, 1, 19, 7, 23, 15, 23, 27, 29, 15, 1, 1, 7, 1, 11, 2, 37, 19, 1, 1, 11, 8, 37, 19, 2, 41, 11, 25, 29, 31, 7, 25, 17, 35, 1, 1, 3, 2, 13, 9, 1, 19, 29, 59

Offset

2,5

Links

Stefano Spezia, Table of n, a(n) for n = 2..10371 (first 1400 rows of the triangle)

Jaba Kalita and Helen K. Saikia, A note on near harmonic divisor number and associated concepts, Palestine Journal of Mathematics, Vol. 13(4), 2024.

Formula

T(n, k) = denominator(n*(tau(n) - 1)/(sigma(n) - n/A027750(n, k))).

Example

The irregular triangle begins as:
  1,  1;
  1,  1;
  3,  5,  3;
  1,  1;
  1,  1,  5, 11;
  1,  1;
  7, 11, 13,  7;
  2,  5,  2;
  4, 13,  8, 17;
  ...
The irregular triangle of the related fractions begins as:
     2,     1;
     3,     1;
   8/3,   8/5,   4/3;
     5,     1;
     3,     2,   9/5,  18/11;
   7,1;
  24/7, 24/11, 24/13,   12/7;
   9/2,   9/5,   3/2;
  15/4, 30/13,  15/8,  30/17;
  ...

Mathematica

T[n_, k_]:=Denominator[n(DivisorSigma[0, n]-1)/(DivisorSigma[1, n]-n/Part[Divisors[n], k])]; Table[T[n, k], {n, 2, 24}, {k, DivisorSigma[0, n]}]//Flatten

Crossrefs

Cf. A000005, A000203, A001599, A027750, A099378, A379945 (numerator).

Sequence in context: A114865 A096196 A351179 * A076824 A103728 A243533

Adjacent sequences: A379943 A379944 A379945 * A379947 A379948 A379949

Keyword

nonn,frac,tabf

Author

Stefano Spezia, Jan 07 2025

Status

approved