A337932 Number of ways to write n as the sum of two deficient numbers (A005100).

0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 6, 8, 6, 7, 6, 7, 7, 9, 7, 9, 8, 9, 8, 11, 8, 10, 9, 11, 10, 13, 10, 12, 11, 13, 11, 15, 11, 13, 12, 15, 13, 18, 13, 15, 14, 17, 15, 20, 15, 17, 15, 18, 16, 22, 16, 19, 17, 20, 18, 24, 18, 22, 19, 22, 19, 25, 19, 22, 20, 25, 21, 28

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{i=1..floor(n/2)} c(i) * c(n-i), where c is the characteristic function of deficient numbers (A294934).

Example

The deficient numbers start: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... .
a(5) = 2; 5 = 4+1 = 3+2 and 1,2,3,4 are all deficient numbers.
a(6) = 3; 6 = 5+1 = 4+2 = 3+3 and 1,2,3,4,5 are all deficient.
a(7) = 2; 7 = 5+2 = 4+3. Since 6 is not a deficient number, the sum 6+1 is not counted.

Maple

N:= 100: # for a(1)..a(N)
S:= select(t -> numtheory:-sigma(t) < 2*t, [$1..N]):
V:= Vector(N): V[S]:= 1:
R:= Vector(N):
for i in select(`<=`, S, N/2) do
  R[2*i..N]:= R[2*i..N] + V[i..N-i]
od:
convert(R, list); # Robert Israel, Oct 09 2020

Mathematica

Table[Sum[(1 - Sign[Floor[DivisorSigma[1, n - i]/(2 (n - i))]])*(1 - Sign[Floor[DivisorSigma[1, i]/(2 i)]]), {i, Floor[n/2]}], {n, 100}]

Crossrefs

Cf. A005100, A294934.

Sequence in context: A108309 A341307 A103469 * A029326 A239500 A029204

Adjacent sequences: A337929 A337930 A337931 * A337933 A337934 A337935

Keyword

nonn,look

Author

Wesley Ivan Hurt, Oct 01 2020

Status

approved