# Curious Proth Primes

This is an ad hoc collection of interesting facts about "Proth primes" of the form k*2^n+1, or of such Proth primes which are curious in some way -- generally by being rare or by holding some sort of record...
First, some definitions:
• k -- on this page, always refers to an odd positive integer
• n -- on this page, always refers to a nonnegative integer
• a -- on this page, always refers to a positive integer
• NSEQ(k) -- given k, the ordered sequence of all n such that k*2^n+1 is prime. For some values of k (known as Sierpinski numbers), this sequence may be empty. The smallest known Sierpinski number is 78557. It is generally believed, but not proven, that for "most" values of k, NSEQ(k) is an infinite sequence.
• KSEQ(n) -- given n, the ordered sequence of all k such that k*2^n+1 is prime. By Dirichlet's Theorem, this sequence is infinite for any value of n.
• NSEQ(k)[a] -- The element of sequence NSEQ(k) at index a, where the index begins with element 1. Undefined if the sequence NSEQ(k) has fewer than a elements.
• KSEQ(n)[a] -- The element of sequence KSEQ(n) at index a, where the index begins with element 1. Always defined, since KSEQ(n) is always an infinite sequence.

## KSEQ(n) == KSEQ(n+1)

For these values of n, we have the curious coincidence that KSEQ(n) == KSEQ(n+1):
• KSEQ(0) == KSEQ(1) == 1
• KSEQ(1) == KSEQ(2) == 1
• KSEQ(5) == KSEQ(6) == 3
• KSEQ(23) == KSEQ(24) == 45
• KSEQ(37) == KSEQ(38) == 15
• KSEQ(42) == KSEQ(43) == 9
• KSEQ(70) == KSEQ(71) == 39
• KSEQ(128) == KSEQ(129) == 21
• KSEQ(555) == KSEQ(556) == 141
• KSEQ(1156) == KSEQ(1157) == 1035
• KSEQ(3971) == KSEQ(3972) == 807
• KSEQ(4532) == KSEQ(4533) == 423

## KSEQ(n)/n >= 5

For these values of n, we have a curious drought of primes for "small" k:
• KSEQ(349) == 1875, ratio == 5.372
• KSEQ(427) == 2375, ratio == 5.562
• KSEQ(1695) == 14319, ratio == 8.448 (!)
• KSEQ(3150) == 15979, ratio == 5.073
• KSEQ(4611) == 25925, ratio == 5.622
• KSEQ(5908) == 30487, ratio == 5.160
Note that for 1 <= n <= 6085, the mean value of KSEQ(n)/n is equal to 0.717, so a ratio of 5 is approximately 7 times larger than the average.