幂数

维基百科，自由的百科全书

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）。

数学性质

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

p为所有的素数
${\displaystyle \zeta (s)}$黎曼ζ函数
${\displaystyle \zeta (3)}$阿培里常数[2]

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

幂数的和与差

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

注解

1. ^ 词都 幂数
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.

延伸阅读

• 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.