A207376 Sum of central divisors of n.

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, 30, 11, 32, 12, 14, 19, 12, 6, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 7, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 8, 18, 17, 68, 21, 26, 17

Offset: 1

Author

Omar E. Pol, Feb 23 2012

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000
• Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Crossrefs

Row sums of A207375. Where records occur give A008578.

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

Programs

Mathematica

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 *)

Formula

a(n) = A000203(n) - A323643(n). - Omar E. Pol, Feb 26 2019