フェルマーへのメルセンヌへの手紙を引用したけれど、約数の和がもとの数の６倍って、半端ないケースであることに本日、気がついた。
どうもメルセンヌの関心は完全数にあったようだ。完全数を復習しておこう。
６は完全数だ。１＋２＋３＝６。２８や４９６もそうだ。
約数をすべて加算するとその数自身になる。ところが完全数となる数は数少ない。
大多数は（約数の和）÷数 つまり、商が分数になってしまう。
偶数の完全数はメルセンヌ素数から算出され、その数だけあるということが証明されている。完全数には、メルセンヌの精神が息づいているのである。
余談はさておき、もとの数とそのすべての約数の和を加えたものをもとの数で割ることをかたはしから、やってみよう。
２〜１０００まではこうなる。２のときは、１＋２＝３となり、商は3/2となる。
3/2,4/3,7/4,6/5,2,8/7,15/8,13/9,9/5,12/11,7/3,14/13,12/7,8/5,31/16,18/17,13/6,20/19,21/10,32/21,18/11,24/23,5/2,31/25,21/13,40/27,2,30/29,12/5,32/31,63/32,16/11,27/17,48/35,91/36,38/37,30/19,56/39,9/4,42/41,16/7,44/43,21/11,26/15,36/23,48/47,31/12,57/49,93/50,24/17,49/26,54/53,20/9,72/55,15/7,80/57,45/29,60/59,14/5,62/61,48/31,104/63,127/64,84/65,24/11,68/67,63/34,32/23,72/35,72/71,65/24,74/73,57/37,124/75,35/19,96/77,28/13,80/79,93/40,121/81,63/41,84/83,8/3,108/85,66/43,40/29,45/22,90/89,13/5,16/13,42/23,128/93,72/47,24/19,21/8,98/97,171/98,52/33,217/100,102/101,36/17,104/103,105/52,64/35,81/53,108/107,70/27,110/109,108/55,152/111,31/14,114/113,40/19,144/115,105/58,14/9,90/59,144/119,3,133/121,93/61,56/41,56/31,156/125,52/21,128/127,255/128,176/129,126/65,132/131,28/11,160/133,102/67,16/9,135/68,138/137,48/23,140/139,12/5,64/47,108/71,168/143,403/144,36/29,111/73,76/49,133/74,150/149,62/25,152/151,75/38,26/17,144/77,192/155,98/39,158/157,120/79,72/53,189/80,192/161,121/54,164/163,147/82,96/55,126/83,168/167,20/7,183/169,162/85,260/171,77/43,174/173,60/29,248/175,93/44,80/59,135/89,180/179,91/30,182/181,24/13,248/183,45/23,228/185,64/31,216/187,84/47,320/189,36/19,192/191,127/48,194/193,147/97,112/65,57/28,198/197,26/11,200/199,93/40,272/201,153/101,240/203,42/17,252/205,156/103,104/69,217/104,240/209,96/35,212/211,189/106,96/71,162/107,264/215,25/9,256/217,165/109,296/219,126/55,252/221,76/37,224/223,9/4,403/225,171/113,228/227,140/57,230/229,216/115,128/77,225/116,234/233,7/3,288/235,105/59,320/237,216/119,240/239,31/10,242/241,399/242,364/243,217/122,342/245,84/41,280/247,60/31,112/83,234/125,252/251,26/9,288/253,192/127,144/85,511/256,258/257,88/43,304/259,147/65,130/87,198/131,264/263,30/11,324/265,240/133,120/89,119/67,270/269,8/3,272/271,279/136,64/39,207/137,372/275,56/23,278/277,210/139,416/279,18/7,282/281,96/47,284/283,126/71,32/19,252/143,48/41,91/32,307/289,54/29,392/291,259/146,294/293,114/49,72/59,285/148,160/99,225/149,336/299,217/75,352/301,228/151,136/101,155/76,372/305,39/17,308/307,24/11,416/309,288/155,312/311,35/13,314/313,237/157,208/105,140/79,318/317,108/53,360/319,381/160,144/107,288/161,360/323,847/324,434/325,246/163,440/327,315/164,384/329,144/55,332/331,147/83,494/333,252/167,408/335,62/21,338/337,549/338,152/113,189/85,384/341,130/57,400/343,165/86,192/115,261/173,348/347,70/29,350/349,372/175,560/351,189/88,354/353,120/59,432/355,315/178,192/119,270/179,360/359,13/4,381/361,273/181,532/363,28/13,444/365,124/61,368/367,93/46,182/123,342/185,432/371,224/93,374/373,324/187,208/125,90/47,420/377,160/63,380/379,42/19,512/381,288/191,384/383,85/32,576/385,291/193,572/387,343/194,390/389,168/65,432/391,855/392,176/131,297/197,96/79,91/33,398/397,300/199,640/399,961/400,402/401,136/67,448/403,357/202,242/135,360/203,456/407,45/17,410/409,378/205,184/137,182/103,480/413,52/23,504/415,441/208,560/417,360/209,420/419,16/5,422/421,318/211,208/141,405/212,558/425,144/71,496/427,189/107,224/143,396/215,432/431,155/54,434/433,384/217,48/29,385/218,480/437,148/73,440/439,27/11,247/147,378/221,444/443,266/111,108/89,336/223,200/149,127/56,450/449,403/150,504/451,399/226,608/453,342/227,96/65,50/19,458/457,345/229,80/51,252/115,462/461,192/77,464/463,465/232,256/155,351/233,468/467,49/18,544/469,432/235,632/471,225/118,48/43,160/79,124/95,36/17,78/53,360/239,480/479,63/20,532/481,363/241,256/161,931/484,588/485,182/81,488/487,465/244,656/489,513/245,492/491,98/41,540/493,420/247,104/55,2,576/497,168/83,500/499,273/125,224/167,378/251,504/503,65/21,612/505,432/253,244/169,224/127,510/509,216/85,592/511,1023/512,800/513,387/257,624/515,308/129,576/517,456/259,232/173,63/26,522/521,65/29,524/523,231/131,992/525,396/263,576/527,31/11,553/529,486/265,260/177,40/19,588/533,180/89,648/535,255/134,240/179,405/269,684/539,28/9,542/541,408/271,728/543,567/272,132/109,32/13,548/547,483/274,806/549,558/275,600/551,60/23,640/553,417/277,304/185,245/139,558/557,208/93,616/559,93/35,288/187,423/281,564/563,112/47,684/565,426/283,968/567,135/71,570/569,48/19,572/571,294/143,256/191,72/41,744/575,1651/576,578/577,921/578,776/579,63/29,96/83,196/97,648/583,555/292,28/15,441/293,588/587,19/7,640/589,108/59,264/197,589/296,594/593,80/33,864/595,525/298,800/597,504/299,600/599,31/10,602/601,528/301,884/603,266/151,798/605,204/101,608/607,315/152,320/203,558/305,672/611,91/34,614/613,462/307,336/205,180/77,618/617,208/103,620/619,336/155,320/207,468/311,720/623,217/78,781/625,471/313,320/209,553/314,684/629,104/35,632/631,150/79,848/633,477/317,768/635,126/53,114/91,540/319,104/71,153/64,642/641,216/107,644/643,48/23,352/215,540/323,648/647,605/216,720/649,651/325,1024/651,287/163,654/653,220/109,792/655,651/328,962/657,576/329,660/659,168/55,662/661,498/331,336/221,315/166,192/133,247/111,720/667,294/167,896/669,612/335,744/671,3,674/673,507/337,248/135,1281/676,678/677,228/113,112/97,81/34,304/227,576/341,684/683,455/171,828/685,600/343,920/687,341/172,756/689,288/115,692/691,609/346,416/231,522/347,168/139,75/29,756/697,525/349,312/233,62/25,702/701,280/117,40/37,381/176,384/235,531/353,816/707,140/59,710/709,648/355,1040/711,675/356,768/713,288/119,1008/715,315/179,320/239,540/359,720/719,403/120,832/721,1143/722,968/723,637/362,186/145,266/121,728/727,30/13,1093/729,666/365,792/731,434/183,734/733,552/367,456/245,189/92,816/737,91/41,740/739,399/185,1120/741,648/371,744/743,80/31,180/149,561/373,364/249,378/187,864/749,312/125,752/751,93/47,336/251,630/377,912/755,80/27,758/757,570/379,384/253,45/19,762/761,256/127,880/763,336/191,156/85,576/383,840/767,511/192,770/769,864/385,344/257,679/386,774/773,286/129,32/25,735/388,1216/777,585/389,840/779,196/65,864/781,648/391,400/261,1767/784,948/785,264/131,788/787,693/394,352/263,144/79,912/791,65/22,868/793,597/397,432/265,350/199,798/797,320/133,864/799,1953/800,130/89,603/401,888/803,476/201,1152/805,672/403,360/269,765/404,810/809,121/45,812/811,60/29,1088/813,684/407,984/815,93/34,880/817,615/409,16/9,441/205,822/821,276/137,824/823,195/103,496/275,720/413,828/827,182/69,830/829,756/415,1112/831,889/416,1026/833,280/139,1008/835,420/209,1280/837,630/419,840/839,24/7,871/841,633/421,376/281,371/211,1098/845,104/47,152/121,837/424,1136/849,837/425,912/851,168/71,854/853,744/427,104/57,405/214,858/857,336/143,860/859,462/215,64/41,648/431,864/863,35/12,1044/865,651/433,1228/867,64/31,960/869,72/29,952/871,825/436,1274/873,720/437,1248/875,518/219,878/877,660/439,392/293,279/110,882/881,247/98,884/883,441/221,96/59,666/443,888/887,95/37,1024/889,162/89,44/27,392/223,960/893,300/149,216/179,255/112,448/299,675/449,960/899,2821/900,972/901,756/451,1408/903,855/452,1092/905,304/151,908/907,399/227,442/303,144/65,912/911,155/57,1008/913,687/457,496/305,805/458,1056/917,40/17,920/919,54/23,1232/921,693/461,1008/923,32/11,1178/925,696/463,1352/927,945/464,930/929,384/155,60/49,819/466,416/311,702/467,1296/935,35/12,938/937,816/469,1256/939,504/235,942/941,316/157,1008/943,465/236,128/63,72/43,948/947,560/237,1036/949,186/95,424/317,270/119,954/953,117/53,1152/955,420/239,480/319,720/479,1104/959,127/40,993/961,798/481,156/107,847/482,1164/965,384/161,968/967,1995/968,480/323,882/485,972/971,637/243,160/139,732/487,1736/975,961/488,978/977,328/163,1080/979,171/70,1430/981,738/491,984/983,105/41,1188/985,810/493,512/329,490/247,1056/989,156/55,992/991,63/31,1328/993,864/497,240/199,196/83,998/997,750/499,1520/999,117/50
これを縦軸にその商、横軸にその数をとるとこうなる。
対象（横軸）を１００万まで延長するとこうなる。水槽のようだ。
商でどのあたりが頻出するかを１００万以下で調べると、1.5あたりが1〜1.1に次いで多い。ただし、1.5となるのは「２」の時だけである。1.6となるのは「１５」だけ、
1.7と1.9は該当なしで、1.8は「１０」だけである．．．というようにかなり粗である。
さて、完全数はさておき、実はその商が自然数となる場合が出てきた。１２０の時に商は３となる。672,523776の時も３だ。30240と32760のときは、商が４になる。だんだん大きい商が出てくる。これがメルセンヌの関心をひいたのではないだろうか？
それどころか、どこからともなく彼は「６」になるものを取り出してきた。これはなんだろう。商自体が完全数だ。
