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%I #26 Sep 18 2024 16:34:03
%S 0,1,1,2,3,5,7,9,10,11,13,17,21,23,23,24,27,31,35,39,41,41,43,49,54,
%T 55,55,57,61,67,73,77,79,79,79,84,91,93,93,97,103,109,115,119,119,121,
%U 123,131,138,141,143,145,149,155,159,163,167,167,169,179,189
%N a(1)=0; for n >= 2, a(n) = a(n-1) + |d(n)-d(n-1)|, where d(n) is the number of positive divisors of n.
%C For any given n >= 2, a(n)/(n-1) is the average of the |d(k)-d(k-1)| over all k with 2 <= k <= n.
%C Partial sums of |A051950|, i.e., a(n) = Sum_{i=2..n} |d(i)-d(i-1)| = Sum_{i=2..n} |A051950(i)|. - _M. F. Hasler_, Apr 21 2008
%H Harvey P. Dale, <a href="/A137377/b137377.txt">Table of n, a(n) for n = 1..1000</a>
%F The following is an empirical formula which is a very good fit for the range n >= 10290 out to about n = 500000000: a(n) ~= n*log(n)+(log(n)*0.122-1)*(n*log(log(n))). - _Jack Brennen_, Apr 21 2008. The constant 0.122 is an empirical guess analogous to Legendre's constant B in Pi(n) ~ n/(log(n)+B).
%t nxt[{n_,a_}]:={n+1,a+Abs[DivisorSigma[0,n+1]-DivisorSigma[0,n]]}; NestList[ nxt,{1,0},60][[All,2]] (* _Harvey P. Dale_, Nov 05 2019 *)
%o (PARI) a(n)=sum(i=2,n,abs(numdiv(i)-numdiv(i-1))) \\ _M. F. Hasler_, Apr 21 2008
%Y Cf. A000005, A051950.
%K nonn
%O 1,4
%A _Leroy Quet_, Apr 21 2008
%E More terms from _M. F. Hasler_, Apr 21 2008
%E Edited by _N. J. A. Sloane_, Apr 26 2008