こんにちは｜こんばんは。カエルのアイコンで活動しております @kyamaz 
です。
はじめに
以前のエントリ名前のついた素数コレクション 〜Haskellを添えて〜で、名前のついた素数について取り上げました。そのエントリでプログラムを紹介しなかった『そのほかの素数』の中から、ラッキー素数（幸運素数）とハッピー素数について詳細を取り上げたいと思います。
なんだか幸せになりそうな名前がついていますが、素数というよりはそれぞれ、ラッキー数とハッピー数という数列があり、その数列のなかで素数のものを言います。本稿では、『ラッキー数』『ハッピー数』について解説し、ハッピーでラッキーな数列とその中から素数であるハッピー＆ラッキー素数をプログラムで求めてみましょう。
ラッキー数
素数列を得る方法のエラトステネスの篩（ふるい）に似た『ウラムの篩』で残った奇数列を「ウラムのラッキー数（幸運数）」といいます。
ウラムの篩
ウラムの篩は次の手順で進めます。
１）1から始まる自然数列から、まずは1を残します
２）次に2を残して、2番目ごとの数を消します（これで奇数が残ります）
３）次に3を残して、上の奇数列の3番目個ごとの数（$5,11,17,\dots$）を消します
４）次に7を残して、上の奇数列の７番目個ごとの数（$19,\dots$)を消します
５）このあと同様に、残っているはじめの数$k$を残して、$k$番目ごとの数を消していきます
ラッキー数列を返すプログラムは次のようになります。
:{
luckyNumbers = 1 : sieve 2 [1,3..] where
sieve k xs = z : sieve (k+1) (lucky xs) where
z = xs !! (k-1)
lucky ws = us ++ lucky vs where
(us, _:vs) = splitAt (z-1) ws
:}
出力のためのchunk関数を使って、100個のラッキー数を出力してみましょう。（chunk関数はこの後でも使います）
ghci> chunk n xs | xs == [] = [] | otherwise = take n xs : chunk n (drop n xs)
ghci> mapM_ print $ chunk 10 $ take 100 luckyNumbers
[1,3,7,9,13,15,21,25,31,33]
[37,43,49,51,63,67,69,73,75,79]
[87,93,99,105,111,115,127,129,133,135]
[141,151,159,163,169,171,189,193,195,201]
[205,211,219,223,231,235,237,241,259,261]
[267,273,283,285,289,297,303,307,319,321]
[327,331,339,349,357,361,367,385,391,393]
[399,409,415,421,427,429,433,451,463,475]
[477,483,487,489,495,511,517,519,529,535]
[537,541,553,559,577,579,583,591,601,613]
ハッピー数
各桁を1桁に分解して二乗和を取り、新しくできた数についても同じ処理を繰り返し行って、最終的に$1$となる数を『ハッピー数』といいます。
定義から、ある数がハッピー数なら、桁の順番を入れ替えた数もハッピー数となります。例えば、$23$はハッピー数なので、$32$はハッピー数です。また、途中に$0$を挿入した数もハッピー数となります。
ハッピー数列を返すプログラムは次のようになります。
import Data.Char (digitToInt)
import Data.Set (member, insert, empty)
isHappy = h empty where h _ 1 = True; h s n | n `member` s = False | otherwise = h (insert n s) (f n) where f = sum . fmap ((^ 2) . toInteger . digitToInt) . show
happyNumber = filter isHappy [1..]
次に、100個のハッピー数を出力してみましょう。
ghci> chunk n xs | xs == [] = [] | otherwise = take n xs : chunk n (drop n xs)
ghci> mapM_ print $ chunk 10 $ take 100 happyNumber
[1,7,10,13,19,23,28,31,32,44]
[49,68,70,79,82,86,91,94,97,100]
[103,109,129,130,133,139,167,176,188,190]
[192,193,203,208,219,226,230,236,239,262]
[263,280,291,293,301,302,310,313,319,320]
[326,329,331,338,356,362,365,367,368,376]
[379,383,386,391,392,397,404,409,440,446]
[464,469,478,487,490,496,536,556,563,565]
[566,608,617,622,623,632,635,637,638,644]
[649,653,655,656,665,671,673,680,683,694]
【ハッピー数に関するトリビア】
- $44488$から続く5つの数は全てハッピー数1
- $7899999999999959999999996$から続く7つの数は全てハッピー数1
- 素数番目の素数のことを超素数と呼びますが、ハッピー数番目のハッピー数のことを超ハッピー数と呼びましょう。そのときの超ハッピー数列は次のようになります。2
超ハッピー数：
$1,28,44,70,97,129,176,192,193,293,319,409,446,563,608, \dots$
ハッピー＆ラッキー数
ハッピー数かつラッキー数であるような自然数を『ハッピー＆ラッキー数』と呼びましょう。
前述のプログラムを使えば、次のように求められます。
happyLucky = [x|x<-luckyNumbers, isHappy (toInteger x)]
ghci> mapM_ print $ chunk 10 $ take 100 happyLucky
[1,7,13,31,49,79,129,133,193,219]
[319,331,367,391,409,487,655,673,739,931]
[937,1009,1029,1039,1093,1209,1233,1251,1275,1281]
[1285,1303,1309,1323,1339,1533,1575,1587,1599,1663]
[1771,1857,1933,1959,1995,2019,2115,2121,2133,2211]
[2221,2257,2511,2527,2545,2557,2571,2575,2715,2725]
[2755,2815,2845,2851,2899,2901,3031,3091,3097,3109]
[3123,3133,3153,3213,3301,3313,3351,3355,3465,3607]
[3661,3709,3763,3789,3879,3897,3931,4069,4255,4257]
[4285,4287,4363,4441,4455,4609,4653,4663,4725,4741]
素数判定関数is_prime'を使って、その中から素数のものを求めましょう。
素数判定
exp_mod a n = iter a (n-1) n where iter a n base | n == 1 = a | even n = (iter a (n `div` 2) base) ^ 2 `mod` base | otherwise = a * (iter a ((n-1) `div` 2) base) ^ 2 `mod` base
is_prime' n | n<=1 = False | otherwise = and $ map (\a -> test n a) $ takeWhile (<n) [2,3,5,7,11,13,17,19] where test n a | gcd n a == 1 = (1 == exp_mod a n) | otherwise = False
次のようになります。
（※ラッキー数の計算には時間を要します。後述の通り、あらかじめラッキー数を求めておくと良い）
ghci> mapM_ print $ chunk 10 $ take 100 $ filter is_prime' luckyHappy
[7,13,31,79,193,331,367,409,487,673]
[739,937,1009,1039,1093,1303,1663,1933,2221,2557]
[2851,3109,3301,3313,3607,3709,3931,4363,4441,4663]
[5527,5569,6163,6367,6373,6661,6733,6763,7603,8233]
[10009,10903,11113,11197,11491,13009,13177,13309,14281,14347]
[14407,14437,14449,14461,15121,16417,17713,18253,18523,18757]
[19333,19963,20899,21787,21841,22381,22651,22921,23893,24181]
[25057,25237,25561,25621,25657,26251,27283,27793,28123,28351]
[28393,28513,28771,28837,28933,30013,30103,30313,30367,30643]
[31033,31039,31699,32803,33091,33289,34603,36037,36067,37117]
おわりに
ウラムの篩のプログラムはそれなりに計算時間がかかります。$40000$までの計算結果を折りたたんで掲載しておきます。このなかに、連続する３つの素数が連続する３つのラッキー数となっているような $3301,3307,3313$ が見つかったりします。眺めて研究してみてはいかがでしょうか。
$40000$までのラッキー数
ghci> mapM_ print $ chunk 10 $ takeWhile (<40000) luckyNumbers
[1,3,7,9,13,15,21,25,31,33]
[37,43,49,51,63,67,69,73,75,79]
[87,93,99,105,111,115,127,129,133,135]
[141,151,159,163,169,171,189,193,195,201]
[205,211,219,223,231,235,237,241,259,261]
[267,273,283,285,289,297,303,307,319,321]
[327,331,339,349,357,361,367,385,391,393]
[399,409,415,421,427,429,433,451,463,475]
[477,483,487,489,495,511,517,519,529,535]
[537,541,553,559,577,579,583,591,601,613]
[615,619,621,631,639,643,645,651,655,673]
[679,685,693,699,717,723,727,729,735,739]
[741,745,769,777,781,787,801,805,819,823]
[831,841,855,867,873,883,885,895,897,903]
[925,927,931,933,937,957,961,975,979,981]
[991,993,997,1009,1011,1021,1023,1029,1039,1041]
[1053,1057,1087,1093,1095,1101,1105,1107,1117,1123]
[1147,1155,1167,1179,1183,1189,1197,1201,1203,1209]
[1219,1231,1233,1245,1249,1251,1261,1263,1275,1281]
[1285,1291,1303,1309,1323,1329,1339,1357,1365,1369]
[1387,1389,1395,1401,1417,1419,1435,1441,1455,1459]
[1471,1473,1485,1491,1495,1497,1501,1503,1519,1533]
[1543,1545,1563,1567,1575,1579,1585,1587,1597,1599]
[1611,1639,1641,1645,1659,1663,1675,1693,1701,1705]
[1711,1723,1731,1737,1749,1765,1767,1771,1773,1777]
[1797,1801,1809,1819,1827,1831,1833,1839,1857,1869]
[1879,1893,1899,1915,1921,1923,1933,1941,1945,1959]
[1963,1965,1977,1983,1987,1995,2001,2019,2023,2031]
[2053,2059,2065,2067,2079,2083,2085,2095,2113,2115]
[2121,2125,2127,2133,2163,2173,2187,2209,2211,2215]
[2217,2221,2239,2251,2253,2257,2271,2277,2281,2283]
[2301,2311,2317,2323,2335,2343,2355,2365,2379,2395]
[2403,2407,2409,2415,2419,2427,2439,2445,2461,2467]
[2473,2479,2491,2493,2505,2511,2523,2527,2545,2557]
[2563,2571,2575,2587,2589,2593,2599,2607,2625,2635]
[2647,2649,2653,2661,2667,2671,2689,2697,2715,2725]
[2755,2763,2773,2781,2785,2787,2797,2815,2817,2821]
[2823,2827,2835,2841,2845,2851,2877,2887,2899,2901]
[2905,2913,2923,2943,2953,2961,2971,2973,2977,2983]
[3003,3007,3027,3031,3037,3039,3049,3055,3073,3075]
[3091,3097,3099,3109,3111,3121,3123,3133,3153,3163]
[3171,3175,3183,3187,3199,3213,3223,3229,3235,3243]
[3259,3261,3289,3297,3301,3307,3313,3325,3339,3351]
[3355,3363,3381,3403,3405,3409,3411,3427,3433,3439]
[3451,3453,3465,3477,3481,3487,3489,3495,3499,3507]
[3559,3565,3571,3579,3595,3597,3603,3607,3613,3621]
[3625,3633,3655,3661,3663,3669,3675,3685,3687,3697]
[3709,3717,3721,3727,3747,3753,3763,3771,3781,3789]
[3793,3795,3811,3843,3847,3849,3865,3873,3879,3889]
[3891,3897,3909,3915,3931,3943,3951,3955,3969,3975]
[3981,3991,3999,4003,4015,4023,4033,4035,4041,4045]
[4063,4069,4081,4095,4105,4107,4129,4131,4161,4165]
[4173,4179,4189,4195,4201,4203,4207,4227,4237,4251]
[4255,4257,4263,4269,4285,4287,4315,4321,4329,4335]
[4363,4377,4383,4389,4399,4413,4431,4441,4443,4455]
[4461,4465,4483,4485,4495,4509,4519,4521,4539,4551]
[4561,4567,4569,4573,4587,4609,4611,4621,4623,4629]
[4645,4647,4653,4663,4671,4675,4695,4699,4713,4717]
[4725,4741,4761,4767,4773,4797,4801,4809,4813,4819]
[4833,4837,4839,4843,4851,4863,4867,4881,4887,4893]
[4929,4951,4963,4965,4969,4977,4987,4989,4993,4999]
[5001,5007,5019,5029,5041,5043,5049,5053,5089,5103]
[5127,5137,5139,5149,5151,5157,5169,5179,5181,5191]
[5211,5217,5229,5233,5235,5253,5259,5277,5283,5293]
[5295,5299,5325,5335,5341,5343,5371,5377,5379,5385]
[5409,5419,5427,5433,5449,5455,5463,5473,5487,5491]
[5503,5515,5527,5547,5551,5559,5569,5577,5587,5589]
[5593,5599,5613,5617,5637,5641,5649,5655,5661,5671]
[5673,5679,5691,5701,5707,5713,5719,5737,5755,5763]
[5767,5769,5803,5809,5817,5827,5833,5839,5851,5869]
[5883,5889,5893,5901,5905,5911,5913,5923,5959,5965]
[5967,5971,5973,5977,5991,5997,6009,6019,6031,6049]
[6055,6061,6079,6093,6111,6115,6123,6141,6147,6159]
[6163,6175,6177,6195,6211,6229,6237,6243,6249,6253]
[6271,6273,6279,6301,6309,6331,6345,6351,6355,6363]
[6367,6369,6373,6379,6385,6399,6411,6415,6427,6433]
[6435,6447,6463,6471,6475,6477,6501,6505,6523,6531]
[6535,6541,6553,6559,6567,6573,6601,6621,6625,6631]
[6661,6663,6667,6669,6679,6687,6693,6715,6723,6733]
[6741,6745,6747,6753,6757,6763,6765,6783,6787,6789]
[6841,6849,6867,6871,6883,6891,6909,6915,6921,6931]
[6933,6937,6951,6981,6985,6999,7003,7009,7035,7041]
[7045,7047,7069,7077,7081,7087,7101,7111,7129,7135]
[7153,7167,7171,7173,7183,7191,7195,7197,7207,7215]
[7231,7233,7237,7245,7249,7279,7293,7299,7311,7321]
[7333,7339,7341,7345,7357,7359,7371,7377,7395,7401]
[7405,7419,7435,7437,7443,7447,7459,7471,7489,7501]
[7503,7507,7531,7533,7549,7551,7563,7567,7575,7585]
[7591,7593,7603,7609,7629,7633,7639,7645,7677,7687]
[7689,7701,7711,7717,7737,7755,7779,7791,7795,7801]
[7803,7813,7827,7833,7837,7855,7881,7885,7897,7899]
[7909,7917,7921,7929,7939,7947,7951,7963,7969,7971]
[8001,8005,8007,8013,8037,8047,8071,8073,8085,8089]
[8107,8109,8127,8131,8139,8151,8161,8169,8173,8175]
[8191,8193,8221,8223,8227,8233,8253,8257,8263,8269]
[8283,8289,8299,8325,8331,8347,8359,8365,8367,8379]
[8409,8413,8421,8445,8449,8467,8473,8475,8479,8487]
[8493,8515,8535,8539,8551,8553,8575,8577,8583,8589]
[8601,8605,8611,8617,8635,8637,8641,8647,8655,8661]
[8673,8683,8719,8757,8769,8773,8787,8793,8805,8809]
[8827,8829,8833,8835,8841,8869,8871,8895,8913,8931]
[8937,8947,8955,8961,8977,8979,8989,8995,8997,9009]
[9031,9033,9061,9063,9073,9075,9081,9085,9115,9117]
[9121,9123,9135,9139,9145,9151,9177,9181,9211,9213]
[9223,9231,9235,9249,9253,9267,9273,9277,9285,9291]
[9303,9315,9331,9339,9349,9373,9387,9399,9403,9409]
[9421,9423,9429,9441,9451,9457,9471,9475,9481,9483]
[9487,9501,9535,9537,9543,9547,9549,9555,9561,9567]
[9613,9621,9625,9631,9643,9649,9661,9663,9675,9687]
[9691,9703,9727,9733,9751,9753,9775,9787,9789,9795]
[9801,9807,9811,9813,9837,9841,9883,9895,9915,9927]
[9937,9957,9961,9979,9985,9987,9997,9999,10003,10009]
[10017,10041,10051,10059,10069,10071,10083,10087,10089,10095]
[10111,10117,10125,10131,10143,10147,10149,10153,10173,10191]
[10195,10197,10221,10239,10255,10261,10275,10279,10291,10311]
[10315,10317,10321,10341,10363,10365,10387,10399,10411,10417]
[10431,10441,10443,10459,10461,10471,10501,10507,10509,10525]
[10531,10533,10555,10563,10569,10575,10585,10587,10597,10599]
[10605,10609,10635,10651,10653,10659,10671,10683,10723,10725]
[10731,10747,10753,10755,10759,10773,10785,10789,10797,10809]
[10843,10849,10873,10881,10891,10903,10909,10911,10921,10923]
[10945,10953,10965,10975,10977,11005,11011,11017,11047,11055]
[11059,11073,11077,11089,11091,11097,11113,11137,11157,11163]
[11173,11181,11193,11197,11205,11217,11223,11227,11239,11245]
[11247,11259,11263,11269,11283,11289,11301,11323,11341,11343]
[11371,11373,11377,11389,11391,11395,11407,11419,11425,11427]
[11437,11439,11475,11479,11487,11491,11521,11529,11535,11539]
[11541,11577,11581,11583,11599,11617,11623,11629,11641,11643]
[11659,11667,11677,11679,11707,11721,11725,11731,11769,11781]
[11797,11803,11811,11815,11823,11827,11833,11835,11845,11847]
[11853,11865,11877,11887,11911,11923,11929,11943,11953,11959]
[11961,11991,11995,12007,12015,12025,12049,12057,12063,12069]
[12079,12097,12103,12117,12121,12133,12139,12141,12151,12159]
[12163,12165,12193,12201,12207,12223,12237,12243,12265,12267]
[12285,12295,12301,12307,12315,12319,12321,12333,12337,12357]
[12363,12373,12379,12391,12399,12411,12417,12427,12435,12457]
[12481,12487,12495,12499,12519,12543,12547,12553,12567,12579]
[12589,12607,12609,12621,12631,12645,12649,12651,12673,12679]
[12699,12709,12711,12729,12735,12741,12759,12763,12777,12781]
[12799,12801,12813,12819,12861,12867,12877,12885,12903,12907]
[12925,12937,12951,12957,12961,12969,12991,12999,13009,13011]
[13021,13027,13029,13053,13057,13063,13077,13083,13113,13117]
[13119,13125,13129,13135,13141,13167,13177,13179,13203,13209]
[13215,13219,13243,13245,13267,13273,13279,13281,13285,13293]
[13309,13317,13323,13327,13335,13345,13347,13359,13363,13371]
[13387,13419,13441,13443,13465,13473,13485,13489,13513,13515]
[13519,13527,13531,13533,13537,13551,13567,13575,13581,13587]
[13599,13609,13633,13657,13677,13681,13693,13695,13707,13723]
[13725,13737,13741,13743,13759,13801,13803,13821,13831,13833]
[13849,13851,13861,13873,13893,13897,13929,13947,13953,13957]
[13995,14007,14017,14031,14035,14037,14041,14049,14053,14059]
[14073,14079,14091,14095,14113,14115,14119,14137,14163,14179]
[14181,14185,14199,14205,14209,14211,14217,14229,14245,14247]
[14253,14281,14287,14293,14307,14313,14317,14331,14347,14349]
[14353,14367,14395,14407,14419,14427,14437,14439,14443,14449]
[14457,14461,14473,14491,14493,14505,14521,14523,14539,14541]
[14565,14587,14595,14599,14601,14607,14625,14661,14667,14689]
[14691,14713,14715,14721,14731,14749,14751,14755,14775,14779]
[14785,14793,14797,14817,14841,14875,14881,14883,14899,14905]
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[33433,33435,33439,33445,33463,33465,33475,33495,33525,33537]
[33541,33549,33561,33567,33595,33601,33603,33615,33633,33645]
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[33813,33835,33837,33843,33855,33861,33885,33895,33897,33907]
[33909,33925,33943,33949,33961,33973,33979,33987,33991,33993]
[34003,34005,34021,34027,34053,34065,34093,34099,34105,34107]
[34113,34125,34129,34131,34161,34171,34183,34191,34197,34201]
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[37939,37941,37957,37959,37963,37969,37989,38001,38031,38035]
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[38757,38769,38775,38779,38787,38791,38797,38799,38811,38815]
[38821,38839,38851,38853,38871,38875,38913,38917,38923,38925]
[38943,38949,38967,38985,38989,39007,39043,39045,39061,39067]
[39073,39093,39115,39123,39135,39139,39159,39171,39177,39193]
[39235,39249,39265,39271,39283,39285,39295,39301,39315,39321]
[39345,39355,39361,39367,39387,39411,39417,39441,39451,39463]
[39475,39481,39507,39513,39553,39567,39571,39573,39579,39585]
[39589,39607,39609,39615,39627,39633,39637,39643,39649,39673]
[39693,39705,39711,39721,39733,39741,39753,39769,39783,39795]
[39801,39825,39831,39837,39841,39849,39861,39865,39883,39889]
[39891,39903,39909,39931,39945,39955,39957,39967,39979,39987]
最後に、本稿を記載するために検証したHaskell環境を記しておきます。お手元の環境で検証する際に、動作が異なるときには参考になるかもしれません。
本稿の環境
本稿のために使用した環境は以下となります。
- macOS: Sonoma 14.5 (chip: Apple M1)
- GHCup: 0.1.30.0
- GHC: 9.6.4
ご一読いただきまして有り難うございます。
（●）（●） Happy Hacking!
/"" ＿＿""＼