ペレリマン数列（2乗するともとに戻る数）の計算をカシオの高精度計算サイトkeisan.casio.jpにUP!(加藤文元さんの天に向かって続く数より）

加藤文元さん、中井保行さん著の天に向かって続く数を読み始めています。

最初から面白いです。表紙も素敵。

で、読み進める前に（ってのも変ですが）、出てくるペレリマン数列の具体的な計算を

カシオの高精度計算サイト、keisan.casio.jpに自作式としてUP!

これです。

ペレリマン数列（2乗してもとにもどる数　天に向かって続く数）

例えばn=10としたら、こんな感じに。n=23までは計算できます。

n	Pn	Qn	Pn+Qn	Pn*Qn
0	5	6	11	30
1	25	76	101	1,900
2	625	376	1,001	235,000
3	625	9,376	10,001	5,860,000
4	90,625	9,376	100,001	849,700,000
5	890,625	109,376	1,000,001	97,413,000,000
6	2,890,625	7,109,376	10,000,001	20,550,540,000,000
7	12,890,625	87,109,376	100,000,001	1,122,894,300,000,000
8	212,890,625	787,109,376	1,000,000,001	167,568,207,000,000,000
9	8,212,890,625	1,787,109,376	10,000,000,001	14,677,333,840,000,000,000
10	18,212,890,625	81,787,109,376	100,000,000,001	1,489,579,677,600,000,000,000

PnとQnを二乗したのはこうなる。

n	Pn	Qn	Pn*Pn	Qn*Qn
0	5	6	25	36
1	25	76	625	5776
2	625	376	390625	141376
3	625	9376	390625	87909376
4	90625	9376	8212890625	87909376
5	890625	109376	793212890625	11963109376
6	2890625	7109376	8355712890625	50543227109376
7	12890625	87109376	166168212890625	7588043387109376
8	212890625	787109376	45322418212890625	619541169787109376
9	8212890625	1787109376	67451572418212890625	3193759921787109376
10	18212890625	81787109376	331709384918212890625	6689131260081787109376

こちらも参照：

ペレリマン数列（2乗するともとに戻る数）を10000桁までPari/GPで計算（天に向かって続く数、加藤文元さん著）

※ちなみに自作式は50桁しか使えませんが、フリー計算にすると130桁まで計算できるので、n=64くらいまで計算できる。

https://keisan.casio.jp/calculator

n	Pn	Qn	Pn+Qn	Pn*Qn
0	5	6	11	30
1	25	76	101	1900
2	625	376	1001	235000
3	625	9376	10001	5860000
4	90625	9376	100001	849700000
5	890625	109376	1000001	97413000000
6	2890625	7109376	10000001	20550540000000
7	12890625	87109376	100000001	1122894300000000
8	212890625	787109376	1000000001	167568207000000000
9	8212890625	1787109376	10000000001	14677333840000000000
10	18212890625	81787109376	100000000001	1489579677600000000000
11	918212890625	81787109376	1000000000001	75097978116000000000000
12	9918212890625	81787109376	10000000000001	811181962500000000000000
13	59918212890625	40081787109376	100000000000001	2401629053056300000000000000
14	259918212890625	740081787109376	1000000000000001	192360735498369000000000000000
15	6259918212890625	3740081787109376	10000000000000001	23412606096826500000000000000000
16	56259918212890625	43740081787109376	100000000000000001	2460813423967920300000000000000000
17	256259918212890625	743740081787109376	1000000000000000001	190590772530413233000000000000000000
18	2256259918212890625	7743740081787109376	10000000000000000001	17471890363594866360000000000000000000
19	92256259918212890625	7743740081787109376	100000000000000000001	714408497724434710200000000000000000000
20	392256259918212890625	607743740081787109376	1000000000000000000001	238391286473188301898000000000000000000000
21	7392256259918212890625	2607743740081787109376	10000000000000000000001	19277109986882123848780000000000000000000000
22	77392256259918212890625	22607743740081787109376	100000000000000000000001	1749664296990971479317600000000000000000000000
23	977392256259918212890625	22607743740081787109376	1000000000000000000000001	22096633663064579877756000000000000000000000000
24	9977392256259918212890625	22607743740081787109376	10000000000000000000000001	225566327323800663862140000000000000000000000000
25	19977392256259918212890625	80022607743740081787109376	100000000000000000000000001	1598643024265518075566205900000000000000000000000000
26	619977392256259918212890625	380022607743740081787109376	1000000000000000000000000001	235605425347387542611699019000000000000000000000000000
27	6619977392256259918212890625	3380022607743740081787109376	10000000000000000000000000001	22375673248578607787973027150000000000000000000000000000
28	6619977392256259918212890625	93380022607743740081787109376	100000000000000000000000000001	618173638551642000427133183400000000000000000000000000000
29	106619977392256259918212890625	893380022607743740081787109376	1000000000000000000000000000001	95252157813131023943176156621000000000000000000000000000000
30	4106619977392256259918212890625	5893380022607743740081787109376	10000000000000000000000000000001	24201872135205387283861389047250000000000000000000000000000000
31	4106619977392256259918212890625	95893380022607743740081787109376	100000000000000000000000000000001	393797670100508450676500549203500000000000000000000000000000000
32	4106619977392256259918212890625	995893380022607743740081787109376	1000000000000000000000000000000001	4089755649753539084602892150766000000000000000000000000000000000
33	9004106619977392256259918212890625	995893380022607743740081787109376	10000000000000000000000000000000001	8967130175853223232745338976135150000000000000000000000000000000000
34	9004106619977392256259918212890625	90995893380022607743740081787109376	100000000000000000000000000000000001	819336725973818526296137978136291400000000000000000000000000000000000
35	109004106619977392256259918212890625	890995893380022607743740081787109376	1000000000000000000000000000000000001	97122211359957993105678080727159729000000000000000000000000000000000000
36	6109004106619977392256259918212890625	3890995893380022607743740081787109376	10000000000000000000000000000000000001	23770109891500025816336898326088487860000000000000000000000000000000000000
37	96109004106619977392256259918212890625	3890995893380022607743740081787109376	100000000000000000000000000000000000001	373959740295702060513273505686928331700000000000000000000000000000000000000
38	896109004106619977392256259918212890625	103890995893380022607743740081787109376	1000000000000000000000000000000000000001	93097656865661717885933891562937904895000000000000000000000000000000000000000
39	896109004106619977392256259918212890625	9103890995893380022607743740081787109376	10000000000000000000000000000000000000001	8158078693825241514416240230826853920520000000000000000000000000000000000000000
40	30896109004106619977392256259918212890625	69103890995893380022607743740081787109376	100000000000000000000000000000000000000001	2135041348817023840836183928028373240639300000000000000000000000000000000000000000
41	230896109004106619977392256259918212890625	769103890995893380022607743740081787109376	1000000000000000000000000000000000000000001	177583095850870333829532312057987479685952000000000000000000000000000000000000000000
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43	23230896109004106619977392256259918212890625	76769103890995893380022607743740081787109376	100000000000000000000000000000000000000000001	1783415076873068521192631321893833652762889100000000000000000000000000000000000000000000
44	423230896109004106619977392256259918212890625	576769103890995893380022607743740081787109376	1000000000000000000000000000000000000000000001	244106504687773479183190370547519825474051952000000000000000000000000000000000000000000000
45	3423230896109004106619977392256259918212890625	6576769103890995893380022607743740081787109376	10000000000000000000000000000000000000000000001	22513799193014785799043122547316299580112723830000000000000000000000000000000000000000000000
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50	557423423230896109004106619977392256259918212890625	442576576769103890995893380022607743740081787109376	1000000000000000000000000000000000000000000000000001	246702550464445381073893140494639621915864692829625000000000000000000000000000000000000000000000000000
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52	557423423230896109004106619977392256259918212890625	99442576576769103890995893380022607743740081787109376	100000000000000000000000000000000000000000000000000001	55431621450323160172480448518256472991647767769001500000000000000000000000000000000000000000000000000000
53	557423423230896109004106619977392256259918212890625	999442576576769103890995893380022607743740081787109376	1000000000000000000000000000000000000000000000000000001	557112702358129658276176406497909503625574159370564000000000000000000000000000000000000000000000000000000
54	1000557423423230896109004106619977392256259918212890625	8999442576576769103890995893380022607743740081787109376	10000000000000000000000000000000000000000000000000000001	9004459076664974402421304922746339655297445001649604940000000000000000000000000000000000000000000000000000000
55	11000557423423230896109004106619977392256259918212890625	88999442576576769103890995893380022607743740081787109376	100000000000000000000000000000000000000000000000000000001	979043478716291137130051592386144757113235639276551948700000000000000000000000000000000000000000000000000000000
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59	863811000557423423230896109004106619977392256259918212890625	136188999442576576769103890995893380022607743740081787109376	1000000000000000000000000000000000000000000000000000000000001	117641555873406453631168038607763205028596412232893155756045000000000000000000000000000000000000000000000000000000000000
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67	66509580863811000557423423230896109004106619977392256259918212890625	33490419136188999442576576769103890995893380022607743740081787109376	100000000000000000000000000000000000000000000000000000000000000000001	2.2274337397012856166649021165743548336878924615638737589969149146254E+135
68	166509580863811000557423423230896109004106619977392256259918212890625	833490419136188999442576576769103890995893380022607743740081787109376	1000000000000000000000000000000000000000000000000000000000000000000001	1.38784140344368986006861298378201631136562526445738453140939663938063E+137
69	2166509580863811000557423423230896109004106619977392256259918212890625	7833490419136188999442576576769103890995893380022607743740081787109376	10000000000000000000000000000000000000000000000000000000000000000000001	1.697133204466342398879397841453268217615709562633269973444053072839119E+139
70	62166509580863811000557423423230896109004106619977392256259918212890625	37833490419136188999442576576769103890995893380022607743740081787109376	100000000000000000000000000000000000000000000000000000000000000000000001	2.3519760446187490939720712757176057989060338970270109320466429843416725E+141

3乗で戻る数はやったが、

３乗すると（奇数回べき乗すると）元に戻る数59161727622001114846846461792218008213239954784512519836425781249（２乗すると元に戻る数はペレリマン数列）

5乗で戻り、かつ驚きの性質のものがあるという！OEISより、https://oeis.org/A068407
897番目は… 7839804103263499879186432

Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).