時事ネタ．(もう流行りは過ぎた．)
なるしす さんの４つの４で１～１００を作ろう(以下URL)が流行っている．よって便乗した．の3回目．
http://www.nicovideo.jp/watch/sm27096518

・4を３つ使ってできる自然数はどんなものがあるか？を調べた．
演算子の数や使用条件などを緩めると表現可能なモデルが大きくなりすぎる．
そのため４を３つだけ使用したときの結果を示そうと考える．

・使える２項演算子は次のようにした．

(x+y)
(x-y)
(x\times y)
(\frac{x}{y})
( x^y )
()

・使える１項演算子は次のようにした．
(x!)(ただし、(x=3,4)の場合に限る)
(\sqrt{x})(ただし、(x>0)の場合に限る)

また、１項演算子の連続した使用は3度までとする．

・４を使用した定数として次のものが使える．
(44,444,4444) (４を並べて使う)
(.4)（小数点を省略）
(.\dot{4})（循環小数）((=\frac{4}{9}))

さらに、計算機の性能の制限が次のようにあるとする．
計算過程において,(1000000)を超えた場合、これを捨てる．

以上の条件で可能な(1000000)以下の正の整数は次の通りである（これ以外にはない）．
計算アルゴリズムを少し変更して、簡単な式がだいたい出てくるようにしました（見やすい）.
[ 1 = {4}^{\left(4-4\right)} ]
[ 2 = \frac{\left(4+4\right)}{4} ]
[ 3 = 4-\frac{4}{4} ]
[ 4 = 4-\left(4-4\right) ]
[ 5 = 4+\frac{4}{4} ]
[ 6 = \frac{4}{.4}-4 ]
[ 7 = \frac{\left(4!+4\right)}{4} ]
[ 8 = \frac{\left(4-.\dot{4}\right)}{.\dot{4}} ]
[ 9 = \frac{\left(4-.4\right)}{.4} ]
[ 10 = \frac{\left(4+.\dot{4}\right)}{.\dot{4}} ]
[ 11 = \frac{44}{4} ]
[ 12 = \left(4+4\right)+4 ]
[ 13 = 4+\frac{4}{.\dot{4}} ]
[ 14 = 4+\frac{4}{.4} ]
[ 15 = 4!-\frac{4}{.\dot{4}} ]
[ 16 = \left(4!-4\right)-4 ]
[ 17 = (最初のない整数　以降はまばら) ]
[ 18 = \frac{\left(4+4\right)}{.\dot{4}} ]
[ 19 = 4!-\frac{\sqrt{4}}{.4} ]
[ 20 = \left(4\times4\right)+4 ]
[ 21 = 4!-\sqrt{\frac{4}{.\dot{4}}} ]
[ 22 = \frac{44}{\sqrt{4}} ]
[ 23 = 4!-\frac{4}{4} ]
[ 24 = \left(4-\left(4-4\right)\right)! ]
[ 25 = \frac{4}{\left(.4\times.4\right)} ]
[ 26 = 4!+\left(4-\sqrt{4}\right) ]
[ 27 = \frac{4!}{\left(.\dot{4}+.\dot{4}\right)} ]
[ 28 = \sqrt{\left(4\times4\right)}+4! ]
[ 29 = \frac{\sqrt{4}}{.4}+4! ]
[ 30 = \frac{4!}{\left(.4+.4\right)} ]
[ 32 = 4\times\left(4+4\right) ]
[ 33 = 4!+\frac{4}{.\dot{4}} ]
[ 34 = \frac{4}{.4}+4! ]
[ 35 = \frac{\left(4!-\sqrt{.\dot{4}}\right)}{\sqrt{.\dot{4}}} ]
[ 36 = \frac{4}{.\dot{4}}\times4 ]
[ 37 = \frac{\left(\sqrt{.\dot{4}}+4!\right)}{\sqrt{.\dot{4}}} ]
[ 38 = \frac{4!}{\sqrt{.\dot{4}}}+\sqrt{4} ]
[ 39 = \frac{\left(4!+\sqrt{4}\right)}{\sqrt{.\dot{4}}} ]
[ 40 = 44-4 ]
[ 42 = 44-\sqrt{4} ]
[ 44 = \sqrt{{\sqrt{44}}^{4}} ]
[ 45 = \frac{\left(4!-4\right)}{.\dot{4}} ]
[ 46 = \sqrt{4}+44 ]
[ 48 = 44+4 ]
[ 50 = \frac{\left(4!-4\right)}{.4} ]
[ 52 = \left(4!+4\right)+4! ]
[ 53 = \frac{\left(4!-.\dot{4}\right)}{.\dot{4}} ]
[ 54 = \frac{\sqrt{\left(4\times4\right)}!}{.\dot{4}} ]
[ 55 = \frac{\left(.\dot{4}+4!\right)}{.\dot{4}} ]
[ 56 = \frac{4!}{.4}-4 ]
[ 58 = 4+\frac{4!}{.\dot{4}} ]
[ 59 = \frac{\left(4!-.4\right)}{.4} ]
[ 60 = \frac{\sqrt{\left(4\times4\right)}!}{.4} ]
[ 61 = \frac{\left(4!+.4\right)}{.4} ]
[ 62 = \sqrt{4}+\frac{4!}{.4} ]
[ 63 = \frac{\left(4!+4\right)}{.\dot{4}} ]
[ 64 = 4\times\left(4\times4\right) ]
[ 65 = \frac{\left(4!+\sqrt{4}\right)}{.4} ]
[ 66 = \frac{44}{\sqrt{.\dot{4}}} ]
[ 68 = 44+4! ]
[ 70 = \frac{\left(4!+4\right)}{.4} ]
[ 72 = \left(4!\times4\right)-4! ]
[ 78 = \frac{4!}{.\dot{4}}+4! ]
[ 80 = 4\times\left(4!-4\right) ]
[ 81 = \sqrt{{\frac{4}{.\dot{4}}}^{4}} ]
[ 84 = \frac{4!}{.4}+4! ]
[ 88 = 44\times\sqrt{4} ]
[ 90 = \frac{4}{\left(.\dot{4}-.4\right)} ]
[ 92 = \left(4!\times4\right)-4 ]
[ 94 = \left(4!\times4\right)-\sqrt{4} ]
[ 96 = \sqrt{\left(4\times4\right)}\times4! ]
[ 98 = \left(4!\times4\right)+\sqrt{4} ]
[ 99 = \frac{44}{.\dot{4}} ]
[ 100 = \left(4!\times4\right)+4 ]
[ 104 = \left(4!+\sqrt{4}\right)\times4 ]
[ 108 = \frac{4!}{.\dot{4}}\times\sqrt{4} ]
[ 110 = \frac{44}{.4} ]
[ 112 = \left(4!+4\right)\times4 ]
[ 120 = \left(4!\times4\right)+4! ]
[ 125 = {\sqrt{\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}}^{4!} ]
[ 128 = \frac{{4}^{4}}{\sqrt{4}} ]
[ 135 = \frac{4!}{\left(.\dot{4}\times.4\right)} ]
[ 144 = 4!\times\left(4+\sqrt{4}\right) ]
[ 150 = \frac{4!}{\left(.4\times.4\right)} ]
[ 160 = \frac{{\sqrt{\sqrt{\sqrt{4}}}}^{4!}}{.4} ]
[ 176 = 4\times44 ]
[ 192 = \left(4+4\right)\times4! ]
[ 216 = \frac{\left(4!\times4\right)}{.\dot{4}} ]
[ 232 = {4}^{4}-4! ]
[ 240 = 4!\times\frac{4}{.4} ]
[ 252 = {4}^{4}-4 ]
[ 254 = {4}^{4}-\sqrt{4} ]
[ 256 = \sqrt{{\left(4\times4\right)}^{4}} ]
[ 258 = \sqrt{4}+{4}^{4} ]
[ 260 = {4}^{4}+4 ]
[ 280 = {4}^{4}+4! ]
[ 288 = \sqrt{\frac{{4!}^{4}}{4}} ]
[ 375 = \frac{4!}{{\sqrt{\sqrt{\sqrt{.4}}}}^{4!}} ]
[ 384 = \left(4!\times4\right)\times4 ]
[ 400 = \sqrt{{\left(4!-4\right)}^{4}} ]
[ 444 = 444 ]
[ 480 = \left(4!-4\right)\times4! ]
[ 484 = \sqrt{{\left(4!-\sqrt{4}\right)}^{4}} ]
[ 512 = {4}^{4}\times\sqrt{4} ]
[ 528 = 4!\times\left(4!-\sqrt{4}\right) ]
[ 540 = \frac{4!}{\left(.\dot{4}-.4\right)} ]
[ 552 = \left(4!\times4!\right)-4! ]
[ 560 = 4!\times\left(4!-\sqrt{.\dot{4}}\right) ]
[ 572 = \left(4!\times4!\right)-4 ]
[ 574 = \left(4!\times4!\right)-\sqrt{4} ]
[ 576 = \frac{{4}^{4}}{.\dot{4}} ]
[ 578 = \sqrt{4}+\left(4!\times4!\right) ]
[ 580 = \left(4!\times4!\right)+4 ]
[ 592 = \left(\sqrt{.\dot{4}}+4!\right)\times4! ]
[ 600 = \left(4!\times4!\right)+4! ]
[ 624 = 4!\times\left(4!+\sqrt{4}\right) ]
[ 625 = {\frac{\sqrt{4}}{.4}}^{4} ]
[ 640 = \frac{{4}^{4}}{.4} ]
[ 672 = 4!\times\left(4!+4\right) ]
[ 676 = {\sqrt{\left(4!+\sqrt{4}\right)}}^{4} ]
[ 729 = {\sqrt{\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}}^{4!} ]
[ 784 = {\sqrt{\left(4!+4\right)}}^{4} ]
[ 864 = \sqrt{\frac{{4!}^{4}}{.\dot{4}}} ]
[ 891 = {4}^{\sqrt{4!}}+\sqrt{\sqrt{.\dot{4}}} ]
[ 1000 = \sqrt{{\sqrt{\sqrt{\frac{4}{.4}}}}^{4!}} ]
[ 1024 = {4}^{4}\times4 ]
[ 1056 = 44\times4! ]
[ 1152 = 4!\times\left(4!\times\sqrt{4}\right) ]
[ 1296 = {\left(4+\sqrt{4}\right)}^{4} ]
[ 1440 = 4!\times\frac{4!}{.4} ]
[ 1536 = {\sqrt{\sqrt{\sqrt{4}}}}^{4!}\times4! ]
[ 1600 = {\frac{4}{\sqrt{.4}}}^{4} ]
[ 1728 = {\sqrt{\sqrt{\sqrt{\frac{4!}{\sqrt{4}}}}}}^{4!} ]
[ 1936 = {\sqrt{44}}^{4} ]
[ 2048 = {\sqrt{\sqrt{\sqrt{4}}}}^{44} ]
[ 2304 = 4!\times\left(4!\times4\right) ]
[ 2916 = {\frac{4!}{.\dot{4}}}^{\sqrt{4}} ]
[ 3600 = {\sqrt{\frac{4!}{.4}}}^{4} ]
[ 4072 = {\sqrt{\sqrt{4}}}^{4!}-4! ]
[ 4092 = {\sqrt{\sqrt{4}}}^{4!}-4 ]
[ 4094 = {\sqrt{\sqrt{4}}}^{4!}-\sqrt{4} ]
[ 4096 = {\left(4+4\right)}^{4} ]
[ 4098 = {\sqrt{\sqrt{4}}}^{4!}+\sqrt{4} ]
[ 4100 = 4+{\sqrt{\sqrt{4}}}^{4!} ]
[ 4120 = {\sqrt{\sqrt{4}}}^{4!}+4! ]
[ 6144 = {4}^{4}\times4! ]
[ 6561 = {\frac{4}{.\dot{4}}}^{4} ]
[ 8000 = {\sqrt{\sqrt{\sqrt{\left(4!-4\right)}}}}^{4!} ]
[ 8192 = {\sqrt{\sqrt{4}}}^{4!}\times\sqrt{4} ]
[ 9216 = {\left(4!\times4\right)}^{\sqrt{4}} ]
[ 10000 = {\frac{4}{.4}}^{4} ]
[ 10240 = \frac{{\sqrt{\sqrt{4}}}^{4!}}{.4} ]
[ 10648 = {\sqrt{\sqrt{\sqrt{\left(4!-\sqrt{4}\right)}}}}^{4!} ]
[ 13824 = \frac{{4!}^{4}}{4!} ]
[ 15625 = {\sqrt{\sqrt{\frac{\sqrt{4}}{.4}}}}^{4!} ]
[ 16384 = 4\times{\sqrt{\sqrt{4}}}^{4!} ]
[ 17576 = {\sqrt{\sqrt{\sqrt{\left(4!+\sqrt{4}\right)}}}}^{4!} ]
[ 20736 = {\frac{4!}{\sqrt{4}}}^{4} ]
[ 21952 = {\sqrt{\sqrt{\sqrt{\left(4!+4\right)}}}}^{4!} ]
[ 22500 = {\frac{\sqrt{4!}}{.4}}^{4} ]
[ 32768 = {\sqrt{\sqrt{\sqrt{4}}}}^{\frac{4!}{.4}} ]
[ 46656 = {\sqrt{\sqrt{\left(4+\sqrt{4}\right)}}}^{4!} ]
[ 64000 = {\sqrt{\sqrt{\frac{4}{\sqrt{.4}}}}}^{4!} ]
[ 65536 = {\left(4\times4\right)}^{4} ]
[ 82944 = \frac{{4!}^{4}}{4} ]
[ 85184 = {\sqrt{\sqrt{\sqrt{44}}}}^{4!} ]
[ 98304 = 4!\times{\sqrt{\sqrt{4}}}^{4!} ]
[ 110592 = {\sqrt{\sqrt{\sqrt{\left(4!\times\sqrt{4}\right)}}}}^{4!} ]
[ 147456 = .\dot{4}\times{4!}^{4} ]
[ 157464 = {\sqrt{\sqrt{\sqrt{\frac{4!}{.\dot{4}}}}}}^{4!} ]
[ 160000 = {\left(4!-4\right)}^{4} ]
[ 165888 = \frac{{4!}^{4}}{\sqrt{4}} ]
[ 216000 = {\sqrt{\sqrt{\sqrt{\frac{4!}{.4}}}}}^{4!} ]
[ 221184 = {4!}^{4}\times\sqrt{.\dot{4}} ]
[ 234256 = {\left(4!-\sqrt{4}\right)}^{4} ]
[ 262144 = {4}^{\frac{4}{.\dot{4}}} ]
[ 331752 = {4!}^{4}-4! ]
[ 331772 = {4!}^{4}-4 ]
[ 331774 = {4!}^{4}-\sqrt{4} ]
[ 331776 = {\sqrt{4!}}^{\left(4+4\right)} ]
[ 331778 = \sqrt{4}+{4!}^{4} ]
[ 331780 = 4+{4!}^{4} ]
[ 331800 = 4!+{4!}^{4} ]
[ 456976 = {\left(4!+\sqrt{4}\right)}^{4} ]
[ 497664 = \frac{{4!}^{4}}{\sqrt{.\dot{4}}} ]
[ 531441 = {\sqrt{\sqrt{\frac{4}{.\dot{4}}}}}^{4!} ]
[ 614656 = {\left(4!+4\right)}^{4} ]
[ 663552 = \sqrt{4}\times{4!}^{4} ]
[ 746496 = \frac{{4!}^{4}}{.\dot{4}} ]
[ 829440 = \frac{{4!}^{4}}{.4} ]
[ 884736 = {\sqrt{\sqrt{\sqrt{\left(4!\times4\right)}}}}^{4!} ]
[ 1000000 = {\sqrt{\sqrt{\frac{4}{.4}}}}^{4!} ]

Posted on: 2015年11月15日, by : UMU