Rekenkundig getal

Een natuurlijk getal $n$ heet een rekenkundig getal als het rekenkundig gemiddelde van zijn delers een geheel getal is.

Het rekenkundig gemiddelde van de delers van $n$ noemt men de rekenkundige functie $A(n)$ :

A(n)={\frac {\sigma (n)}{d(n)}}

Hierin is $\sigma (n)$ de som van alle positieve delers van $n$ en $d(n)$ het aantal positieve delers van $n$ . Als $A(n)$ een geheel getal is, dus als $d(n)$ een deler is van $\sigma (n)$ , heet $n$ een rekenkundig getal.

Voorbeeld: 14 heeft als delers 1, 2, 7 en 14. Het rekenkundig gemiddelde daarvan is (1+2+7+14)/4 = 6, dus 14 is een rekenkundig getal. Het getal 12 is geen rekenkundig getal, want de som van de delers van 12 is 1+2+3+4+6+12 = 28 en het gemiddelde 28/6 is geen geheel getal.

De eerste rekenkundige getallen zijn:

1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ...^[1]

De functie $A(n)$ is een multiplicatieve functie. Immers $\sigma (n)$ en $d(n)$ zijn beide multiplicatieve functies. Hieruit volgt dat als twee rekenkundige getallen relatief priem zijn, hun product ook een rekenkundig getal is.

Elk oneven priemgetal $p$ is een rekenkundig getal; immers de delers ervan zijn 1 en $p$ , en $A(p)=(p+1)/2$ is een geheel getal omdat $p+1$ een even getal is. 2 is geen rekenkundig getal en ook geen enkele macht van 2 is een rekenkundig getal.^[2]

De asymptotische dichtheid van de verzameling van rekenkundige getallen is gelijk aan 1.^[3]

Voor elk getal $N$ bestaat er een geheel getal $m$ , zodanig datde vergelijking $A(n)=m$ ten minste $N$ oplossingen heeft.^[3]

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