A207376 - OEIS

%I #51 Feb 26 2019 09:13:33

%S 1,3,4,2,6,5,8,6,3,7,12,7,14,9,8,4,18,9,20,9,10,13,24,10,5,15,12,11,

%T 30,11,32,12,14,19,12,6,38,21,16,13,42,13,44,15,14,25,48,14,7,15,20,

%U 17,54,15,16,15,22,31,60,16,62,33,16,8,18,17,68,21,26,17

%N Sum of central divisors of n.

%C If n is a square (A000290) then a(n) = sqrt(n) because the squares have only one central divisor. If n is a prime p then a(n) = 1 + p = A000203(n). For the number of central divisors of n see A169695.

%H Alois P. Heinz, <a href="/A207376/b207376.txt">Table of n, a(n) for n = 1..5000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of the divisors of the first 12 positive integers</a>

%F a(n) = A000203(n) - A323643(n). - _Omar E. Pol_, Feb 26 2019

%e For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12. The central (or middle) divisors of 12 are 3 and 4, so a(12) = 3 + 4 = 7.

%t cdn[n_]:=Module[{dn=Divisors[n],len},len=Length[dn]; Which[ IntegerQ[ Sqrt[n]], Sqrt[n], PrimeQ[n],n+1, OddQ[len],dn[[Floor[len/2]+1]], EvenQ[len],dn[[len/2]]+dn[[len/2+1]]]]; Array[cdn,70] (* _Harvey P. Dale_, Nov 07 2012 *)

%Y Row sums of A207375. Where records occur give A008578.

%Y Cf. A000005, A000040, A000203, A000290, A027750, A161840, A169695, A323643.

%K nonn,easy

%O 1,2

%A _Omar E. Pol_, Feb 23 2012