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

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

##
KSEQ(n)[1]/n >= 5

For these values of **n**, we have a curious drought of primes for "small" **k**:
**KSEQ(349)[1]** == **1875**, ratio == 5.372
**KSEQ(427)[1]** == **2375**, ratio == 5.562
**KSEQ(1695)[1]** == **14319**, ratio == 8.448 (!)
**KSEQ(3150)[1]** == **15979**, ratio == 5.073
**KSEQ(4611)[1]** == **25925**, ratio == 5.622
**KSEQ(5908)[1]** == **30487**, ratio == 5.160

Note that for 1 <= **n** <= 6085, the mean value of KSEQ(n)[1]/n is equal to 0.717, so a ratio of 5 is approximately 7 times larger than the average.