A379945 Irregular triangle read by rows: T(n, k) is the numerator of the harmonic mean of all positive divisors of n except the k-th of them.

2, 1, 3, 1, 8, 8, 4, 5, 1, 3, 2, 9, 18, 7, 1, 24, 24, 24, 12, 9, 9, 3, 15, 30, 15, 30, 11, 1, 15, 30, 5, 12, 30, 20, 13, 1, 21, 42, 21, 42, 5, 45, 15, 45, 64, 64, 64, 64, 32, 17, 1, 30, 3, 30, 5, 90, 45, 19, 1, 50, 25, 100, 50, 5, 100, 63, 63, 63, 63, 33, 66, 33, 66, 23, 1

Offset

2,1

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

Example

The irregular triangle begins as:
   2,  1;
   3,  1;
   8,  8,  4;
   5,  1;
   3,  2,  9, 18;
   7,  1;
  24, 24, 24, 12;
   9,  9,  3;
  15, 30, 15, 30;
  ...
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_]:=Numerator[n(DivisorSigma[0, n]-1)/(DivisorSigma[1, n]-n/Part[Divisors[n], k])]; Table[T[n, k], {n, 2, 23}, {k, DivisorSigma[0, n]}]//Flatten

Crossrefs

Cf. A000005, A000203, A001599, A027750, A099377, A379946 (denominator).

Sequence in context: A373986 A136179 A185176 * A198315 A126761 A090559

Adjacent sequences: A379942 A379943 A379944 * A379946 A379947 A379948

Keyword

nonn,frac,tabf

Author

Stefano Spezia, Jan 07 2025

Status

approved