cp's OEIS Frontend

A172433 Let u(n) = Sum [n/[sqrt k]] and v(n) = Sum [n/(sqrt k)] where the summation index k ranges from 1 to infinity, although both sums are actually finite. Here [a] denotes the integer part of a. Then a(n) = u(n) - v(n).

%I A172433 #10 Jun 25 2021 23:14:58
%S A172433 2,6,9,16,17,27,26,36,38,48,43,67,59,67,72,88,75,102,86,111,115,123,
%T A172433 99,150,137,142,139,169,141,192,166,192,186,189,176,253,214,217,207,
%U A172433 263,223,284,239,269,285,285,230,332,294,325,305,339,282,350,324,391,370,369,300,448,382,377,385,438,400
%N A172433 Let u(n) = Sum [n/[sqrt k]] and v(n) = Sum [n/(sqrt k)] where the summation index k ranges from 1 to infinity, although both sums are actually finite. Here [a] denotes the integer part of a. Then a(n) = u(n) - v(n).
%C A172433 One can pick out the values of the sequence at primes, obtaining the new sequence 6,9,17,26,43,59,75,86,99,141 which seems to be monotone, unlike the original sequence.
%C A172433 Actually, the infinite sum can be replaced by a finite sum with terms up to (n+1)^2 (see second PARI script). Apparently v(n) is A153818(n). - _Michel Marcus_, Jul 17 2013
%o A172433 (PARI) a(n) = round(suminf(k=1, floor(n/sqrtint(k))) - suminf(k=1, floor(n/sqrt(k)))) \\ _Michel Marcus_, Jul 17 2013
%o A172433 (PARI) a(n) = sum(k=1, (n+1)^2, floor(n/sqrtint(k))) - sum(k=1, (n+1)^2, floor(n/sqrt(k))) \\ _Michel Marcus_, Jul 17 2013
%K A172433 nonn
%O A172433 1,1
%A A172433 Ali A. Tanara (aatanara(AT)gmail.com), Feb 02 2010
%E A172433 Definition clarified by Gihan Marasingha (G_Marasingha(AT)hotmail.com), Feb 10 2010