- First I give an equivalent definition of a Cunningham chain:

- A Cunningham chain of the first kind (second kind) is a sequence

of integers of the form k*2^n-1 (k*2^n+1) where k is a fixed

positive odd integer, and n ranges over two or more consecutive

nonnegative integers, and where each member of the sequence has

exactly 2 positive divisors.

You'll note this is a pretty standard definition of a Cunningham

chain, except I've defined it in terms of number of divisors

rather than primality. Why? Because I'm looking at an extension

of Cunningham chains where all of the members of the sequence have

an equal number of positive divisors, but that number is not 2.

So far, I've found a "4-divisor Cunningham chain" (of the second kind)

of length 18:

80935905*2^13+1 == 254993 * 2600177

80935905*2^14+1 == 1951 * 679679071

80935905*2^15+1 == 71 * 37353630071

80935905*2^16+1 == 11 * 482201406371

80935905*2^17+1 == 15667 * 677119483

80935905*2^18+1 == 389 * 54542061389

80935905*2^19+1 == 19 * 2233353882139

80935905*2^20+1 == 4133 * 20534102957

80935905*2^21+1 == 2797 * 60684624613

80935905*2^22+1 == 2181271 * 155629351

80935905*2^23+1 == 13 * 52226121551557

80935905*2^24+1 == 853 * 1591886471677

80935905*2^25+1 == 1489 * 1823880672049

80935905*2^26+1 == 11 * 493774240123811

80935905*2^27+1 == 44287967 * 245281823

80935905*2^28+1 == 689383 * 31515234007

80935905*2^29+1 == 76446859 * 568396579

80935905*2^30+1 == 28879 * 3009254692399

It looks like chains with 4 divisors each seem

to be easiest to find, but 8 divisors each might be worth looking

into as well. The best "8-divisor Cunningham chain" I've found,

without spending a lot of time on it:

302457*2^13+1 == 5 * 97 * 5108717

302457*2^14+1 == 59 * 83 * 1011937

302457*2^15+1 == 11 * 12041 * 74827

302457*2^16+1 == 19 * 1993 * 523459

302457*2^17+1 == 5 * 1129 * 7022789

302457*2^18+1 == 23 * 2521 * 1367423

302457*2^19+1 == 37 * 71 * 60363371

302457*2^20+1 == 73 * 373 * 11647477

302457*2^21+1 == 5 * 329267 * 385279

302457*2^22+1 == 337 * 11273 * 333929

302457*2^23+1 == 67 * 119677 * 316423

302457*2^24+1 == 13 * 251 * 1555129151

Can anybody find any longer such chains each with an equal number

of divisors, either with 4, 8, or some other number?

Jack - I previously wrote:
> So far, I've found a "4-divisor Cunningham chain" (of the second kind)

Add to that a "4-divisor Cunningham chain" (of the first kind) of length 18:

> of length 18...

899643225*2^8+1 == 173 * 1331263963

899643225*2^9+1 == 2347 * 196257917

899643225*2^10+1 == 53 * 17381786083

899643225*2^11+1 == 397 * 4640980667

899643225*2^12+1 == 821 * 4488354019

899643225*2^13+1 == 2351 * 3134784049

899643225*2^14+1 == 1531 * 9627534029

899643225*2^15+1 == 19 * 1551553115621

899643225*2^16+1 == 71 * 830408709769

899643225*2^17+1 == 11 * 10719821526109

899643225*2^18+1 == 193 * 1221948567743

899643225*2^19+1 == 101 * 4670021258899

899643225*2^20+1 == 532663 * 1770996473

899643225*2^21+1 == 499 * 3780939055301

899643225*2^22+1 == 9008023 * 418890713

899643225*2^23+1 == 19427 * 388467306037

899643225*2^24+1 == 122953 * 122758360583

899643225*2^25+1 == 5205467 * 5799098797

So I've got two examples of length 18, a "4DCC1K" and a "4DCC2K". :)

The challenge is this: find a chain of length 19, any number

of divisors you wish. :)