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