Samoštevilo

Sámoštevílo ali Kolumbijevo število je v matematiki pozitivno celo število, ki ga v dani osnovi ne moremo tvoriti z nekim drugim celim številom, seštetim s svojimi števkami. Število 21, na primer, ni samoštevilo, ker ga lahko dobimo iz vsote števila 15 in njegovih števk, oziroma 21 = 15 + 1 + 5. Za število 20 takšna vsota ne obstaja in zato je 20 samoštevilo. Samoštevila je prvi raziskoval leta 1949 indijski matematik Šri Datatreja Ramačandra Kaprekar (1905-1986). Prva samoštevila v desetiškem sestavu so (OEIS A003052):

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525, ...

Pri sodih osnovah so v splošnem vsa soda števila, manjša od osnove, samoštevila, ker mora vsako število manjše od lihega števila biti tudi enomestno in, če ga seštejemo z njego števko, da liho število. Za lihe osnove so vsa liha števila samoštevila.

Definicija in lastnosti

Naj bo $n$ naravno število. Definiramo $b$ -samofunkcijo za osnovo $b>1$ $F_{b}:\mathbb {N} \rightarrow \mathbb {N}$ :

F_{b}(n)=n+\sum _{i=0}^{k-1}d_{i}.

kjer je $k=\lfloor \log _{b}{n}\rfloor +1$ število števk v številki v osnovi $b$ , in

d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}

je vrednost vsake števke števila. Naravno število $n$ je $b$ -samoštevilo če je predpreslikava $n$ za $F_{b}$ prazna množica.

Na splošno velja, tudi za osnove, da so vsa liha števila samoštevila, saj bi katerokoli število, manjše od tega sodega števila, moralo biti 1-števčno število, ki kadar je dodano svoji števki tvori sodo število. Za lihe baze so vsa liha števila samoštevila.^[1]

Množica samoštevil v dani množici $b$ je neskončna in ima pozitivno asimptotično gostoto: ko je $b$ lih, je ta gostota 1/2.^[2]

Rekurenčna enačba

Naslednja rekurenčna enačba daje samoštevila za osnovo 10:

C_{k}=8\cdot 10^{k-1}+C_{k-1}+8\!\,,

kjer je C₁ = 9 in k > 1.

Podobno za dvojiška števila:

C_{k}=2^{j}+C_{k-1}+1\!\,,

kjer j teče po številu števk, C₁ = 1 in k > 0. To rekurenčno enačbo lahko posplošimo za samoštevila v poljubni osnovi b:

C_{k}=(b-2)b^{k-1}+C_{k-1}+(b-2)\!\,,

kjer je C₁ = b - 1 za sode osnove in C₁ = b - 2 za lihe, ter k > 1.

Te rekurenčne enačbe kažejo, da v poljubni osnovi obstaja neskončno mnogo samoštevil.

Obstajajo tudi praštevilska samoštevila. Prva so (OEIS A006378):

3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, ...

Sklici

↑ Sándor & Crstici (2004) p.384
↑ Sándor & Crstici (2004) p.385

Kaprekar, D. R. The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20.
R. B. Patel (1991). »Some Tests for k-Self Numbers«. Math. Student. 56: 206–210.
B. Recaman (1974). »Problem E2408«. Amer. Math. Monthly. 81 (4): 407. doi:10.2307/2319017.
Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. str. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.
Weisstein, Eric Wolfgang. »Self Number«. MathWorld.

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