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