cp's OEIS Frontend

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.

%I A379945 #14 Jan 09 2025 19:18:50
%S A379945 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,
%T A379945 30,5,12,30,20,13,1,21,42,21,42,5,45,15,45,64,64,64,64,32,17,1,30,3,
%U A379945 30,5,90,45,19,1,50,25,100,50,5,100,63,63,63,63,33,66,33,66,23,1
%N 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.
%H A379945 Stefano Spezia, <a href="/A379945/b379945.txt">Table of n, a(n) for n = 2..10371</a> (first 1400 rows of the triangle)
%H A379945 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 A379945 T(n, k) = numerator(n*(tau(n) - 1)/(sigma(n) - n/A027750(n, k))).
%e A379945 The irregular triangle begins as:
%e A379945    2,  1;
%e A379945    3,  1;
%e A379945    8,  8,  4;
%e A379945    5,  1;
%e A379945    3,  2,  9, 18;
%e A379945    7,  1;
%e A379945   24, 24, 24, 12;
%e A379945    9,  9,  3;
%e A379945   15, 30, 15, 30;
%e A379945   ...
%e A379945 The irregular triangle of the related fractions begins as:
%e A379945      2,     1;
%e A379945      3,     1;
%e A379945    8/3,   8/5,   4/3;
%e A379945      5,     1;
%e A379945      3,     2,   9/5,  18/11;
%e A379945    7,1;
%e A379945   24/7, 24/11, 24/13,   12/7;
%e A379945    9/2,   9/5,   3/2;
%e A379945   15/4, 30/13,  15/8,  30/17;
%e A379945   ...
%t A379945 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
%Y A379945 Cf. A000005, A000203, A001599, A027750, A099377, A379946 (denominator).
%K A379945 nonn,frac,tabf
%O A379945 2,1
%A A379945 _Stefano Spezia_, Jan 07 2025