## 15543Extension of Cunningham chains

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• Nov 12, 2004
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
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