A361879 Sum of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 4, 0, 6, 0, 4, 0, 0, 0, 6, 0, 0, 0, 8, 0, 6, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 6, 0, 8, 0, 0, 0, 16, 0, 0, 0, 8, 0, 6, 0, 0, 0, 10, 0, 14, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 12, 0, 0, 0, 8, 0, 10, 0, 0, 0, 0, 0, 20

Offset: 1

Author

Omar E. Pol, Mar 27 2023

Keywords

• nonn
• look

Comments

Sum of even divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also sum of even numbers in the n-th row of A299761.

Examples

			For n = 18 the middle divisor of 18 is [3]. There are no even middle divisors of 18 so a(18) = 0.
For n = 20 the middle divisors of 20 are [4, 5]. There is only one even middle divisor of 20 so a(20) = 4.
For n = 24 the middle divisors of 24 are [4, 6]. There are two even middle divisors of 24 so a(24) = 4 + 6 = 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^16 (ignoring zeros).

Crossrefs

Cf. A000203, A067742, A071090, A071562, A146076, A299761, A299777, A303297, A358434, A361561, A361824.

Programs

Maple

f:= proc(n) local D;
     if n::odd then return 0 fi;
     D:= select(proc(d) local s; if d::odd then return false fi; s:= d^2; s >= n/2 and s < 2*n end proc, numtheory:-divisors(n)); convert(D,`+`) end proc:
map(f, [$1..100]); # Robert Israel, Mar 18 2024

Mathematica

Table[DivisorSum[n, # &, And[EvenQ[#], Sqrt[n/2] <= # < Sqrt[2 n]] &], {n, 120}] (* Michael De Vlieger, Mar 28 2023 *)

PARI

a(n) = vecsum(select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2)) && !(x%2)), divisors(n))); \\ Michel Marcus, Mar 31 2023

Formula

a(n) = A071090(n) - A361824(n).