# 冪數

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000 （OEIS中的数列A001694）。

## 數學性質

$\prod_p(1+\frac{1}{p(p-1)})=\frac{\zeta(2)\zeta(3)}{\zeta(6)}=\frac{315}{2\pi^4}\zeta(3)$

p為所有的質數
$\zeta(s)$黎曼ζ函數
$\zeta(3)$阿培里常數[2]

$cx^{1/2}-3x^{1/3}\le k(x) \le cx^{1/2}, c=\zeta(3/2)/\zeta(3)=2.173\cdots$[2]

### 冪數的和與差

2=33-52
10=133-37
18=192-73=32(33-52)

6=5473-4632

## 一般化

(2k+1}-1)k, 2k(2k+1-1)k, (2k+1-1)k+1

a1(as+d)k, a2(as+d)k, ..., as(as+d)k, (as+d)k+1

ak(an+...+1)k+ak+1(an+...+1)k+...+ak+n(an+...+1)k=ak(an+...+1)k+1

X=9712247684771506604963490444281, Y=32295800804958334401937923416351, Z=27474621855216870941749052236511

## 註解

1. ^ 詞都 幂数
2. ^ 2.0 2.1 2.2 S. W. Golomb, Powerful numbes, Amer. Math. Monthly 77(1970), 848--852.
3. ^ Wayne L. McDaniel, Representations of every integer as the difference of powerful numbers, Fibonacci Quart. 20(1982), 85--87.
4. ^ D. R. Heath-Brown, Sums of three square-full numbers, in Number Theory, I(Budapest, 1987), Colloq. Math. Soc. János Bolyai 51(1990), 163--171. Brown, 1987)
5. ^ *A. Nitaj, On a conjecture of Erdös on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), 317--318.

## 延伸閱讀

• J. H. E. Cohn, A conjecture of Erdös on 3-powerful numbers, Math. Comp. 67 (1998), 439--440. [1]
• P. Erdös & G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Litt. Sci. Szeged 7(1934), 95--102.
• Richard Guy, Section B16 in Unsolved Problems in Number Theory, Springer-Verlag, 3rd edition, 2004; ISBN 0-387-20860-7.
• D. R. Heath-Brown, Ternary quadratic forms and sums of three square-full numbers, Séminaire de Théorie des Nombres, Paris, 1986-7, Birkhäuser, Boston, 1988.