# Mutually Coprime Sequences

I use the term "mutually co-prime sequence" to refer to any sequence of integers for which all terms (possibly excluding a small finite number of initial terms) are pairwise mutually co-prime. This means you can take any two terms (again, possibly with a few exceptions) and they will have no common factors.

Many such sequences exist (for example, any subset of the prime numbers will do) but it gets interesting when the sequence can be generated by a simple formula.

The simplest way to define such a sequence by formula is to define each term as the product of all the previous terms, plus or minus 1. This method generates each of the following sequences, with the only difference being the choice of initial terms:

2, 3, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608869, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068029, ...

A110389 : product of all preceding terms, minus 1. Note the recurrence of the digit endings 029 and 869.

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...

A0058 "Sylvester's sequence": product of all
preceding terms, plus 1. Also, A_{n+1} = A_{n}^{2} - A_{n} + 1.
Again, note the recurrence of the digit endings, this time 807 and 443.

2, 3, 7, 13, 97, 193, 18817, 37633, 708158977, 1416317953, 1002978273411373057, ...

A6695 "An infinite coprime sequence defined by recursion." (%%% came up during the 2014 OEIS conference)

2, 5, 9, 89, 8009, 64152089, 4115490587216009, 16937262773463574696951813104089, ...

Product of all preceding terms, minus 1. (like A110389, but starting with 2 and 5)

2, 5, 11, 111, 12211, 149096311, 22229709804712410, 494159998001727075769152612720511, ...

Product of all preceding terms, plus 1. (like A000058, but starting with 2 and 5)

3, 2, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, ...

A5267 : same as A110389 except for the initial two terms.

3, 7, 47, 2207, 4870847, 23725150497407, 562882766124611619513723647, ...

A1566 : Previous term squared minus 2. Also,
A_{n}+1 = 4 Product_{for all i<n}[A_{i}-1]. See 47
for a discussion of why this sequence is mutually co-prime.

### Pairwise Coprime

Some sequences only have the co-prime property between adjacent terms. For example:

0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, ...

A3095 : A_{n+1} = A_{n}^{2} + 1.

Some other sequences are discussed here.

Notes for NJAS:

- Two of these sequences do not exist in OEIS.

- Note my formula for A001566 (which might or might not demonstrate the mutually coprime property, I'm not sure!)

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