A379946 - OEIS

%I #13 Jan 09 2025 19:18:59

%S 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,

%T 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,

%U 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

%N 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.

%H Stefano Spezia, <a href="/A379946/b379946.txt">Table of n, a(n) for n = 2..10371</a> (first 1400 rows of the triangle)

%H Jaba Kalita and Helen K. Saikia, <a href="https://pjm.ppu.edu/paper/1884-note-near-harmonic-divisor-number-and-associated-concepts">A note on near harmonic divisor number and associated concepts</a>, Palestine Journal of Mathematics, Vol. 13(4), 2024.

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

%e The irregular triangle begins as:

%e   1,  1;

%e   1,  1;

%e   3,  5,  3;

%e   1,  1;

%e   1,  1,  5, 11;

%e   1,  1;

%e   7, 11, 13,  7;

%e   2,  5,  2;

%e   4, 13,  8, 17;

%e   ...

%e The irregular triangle of the related fractions begins as:

%e      2,     1;

%e      3,     1;

%e    8/3,   8/5,   4/3;

%e      5,     1;

%e      3,     2,   9/5,  18/11;

%e    7,1;

%e   24/7, 24/11, 24/13,   12/7;

%e    9/2,   9/5,   3/2;

%e   15/4, 30/13,  15/8,  30/17;

%e   ...

%t 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

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

%K nonn,frac,tabf

%O 2,5

%A _Stefano Spezia_, Jan 07 2025