A283763 Number of ways of writing n as sum of a deficient number (A005100) and abundant number (A005101).

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

Offset

0,20

Links

Table of n, a(n) for n=0..95.

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Abundant Number

Eric Weisstein's World of Mathematics, Deficient Number

Formula

G.f.: (Sum_{i>=1} x^A005100(i))*(Sum_{j>=1} x^A005101(j)).

Example

a(21) = 3 because we have [20, 1], [18, 3] and [12, 7].

Mathematica

nmax = 95; CoefficientList[Series[Sum[Boole[DivisorSigma[1, k] > 2 k] x^k, {k, 1, nmax}] Sum[Boole[DivisorSigma[1, k] < 2 k] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=1, 95, (sigma(k)>2*k) * x^k) * sum(k=1, 95, (sigma(k)<2*k) * x^k) + O(x^96))) \\ Indranil Ghosh, Mar 16 2017

Crossrefs

Cf. A005100, A005101.

Sequence in context: A131619 A048485 A127714 * A357622 A357621 A220604

Adjacent sequences: A283760 A283761 A283762 * A283764 A283765 A283766

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 16 2017

Status

approved