清一色数字中的素数和余因子

yangchuanju

A040017给出47个一路增大的唯一φ因子具体数值（30以后略）：
1  3
2  11
3  37
4  101
5  9091
6  9901
7  333667
8  909091
9  99990001
10  999999000001
11  9999999900000001
12  909090909090909091
13  1111111111111111111
14  11111111111111111111111
15  900900900900990990990991
16  909090909090909090909090909091
17  100009999999899989999000000010001
18  10000099999999989999899999000000000100001
19  9090909090909090909090909090909090909090909090909091
20  900900900900900900900900900900990990990990990990990990990991
21  909090909090909090909090909090909090909090909090909090909090909091
22  142857157142857142856999999985714285714285857142857142855714285571428571428572857143
23  999999999999990000000000000099999999999999000000000000009999999999999900000000000001
24  99999999999999999999999999999999000000000000000000000000000000009999999999999999999999999999999900000000000000000000000000000001
25  1098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098900989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989011
26  1000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001001
27  1111099888878889001111100000000000011110998888788890011222109988887889011221109998887788901111110999878890112222210987766677901122222109888790001111110988877889000111221098887888900112221099888878889001111100000000000011110998888788890011111
28  9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091
29  11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
30  900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990991

yangchuanju

太阳先生不是热衷于研究大素数吗？
下面是第45,46,47号清一色数的唯一φ因子，分别有1001位，1031位，1128位：
45  10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001——1001位，对应清一色数是R3750

46  11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111——1031位，对应清一色数是R1031

47  999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990001——1128位，对应清一色数是R2264

yangchuanju

清一色数的φ因子与素数倒数的循环节
10进制中，每一个素数的倒数都有一个固定的循环节长度，其最大长度等于素数值减1，或者是素数值减1的某个分数。
例7,17,19的倒数循环节长度分别是6,16,18，为最大循环节长，称这些素数都有原根10；
又例素数11,13,37的倒数循环节长分别为2,6,3，都不到最大循环节长度值，分别为最大循环节长的1/5,1/2,1/12，它们都没有原根10。

进一步探讨表明，素数7,17,19分别为清一色数R6，R16，R18的φ因子，其循环节长度分别为6,16,18；
素数11,13,37分别是清一色数R6，R2，R3的φ因子，其循环节长度分别为2,6,3；
综上所述，清一色数字的φ因子素数其循环节长度就等于清一色数字的指数！

循环节长度等于1的只有一个素数3；循环节长度等于2的只有一个素数11；循环节长度等于3的只有一个素数37；循环节长度等于4的只有一个素数101；
循环节长度等于5的有两个素数41和271；循环节长度等于6的有两个素数7和13；循环节长度等于7的有两个素数239和4649；循环节长度等于8的有两个素数73和137；
循环节长度等于9的有一个素数333667；循环节长度等于10的有一个素数9091；循环节长度等于11的有两个素数21649和513239；循环节长度等于12的有一个素数9901；
……

某清一色数字Rn若有k个φ因子，则以n位循环节长度的素数就有k个！