[seqfan] Two questions looking for answers

[Neil Sloane]

Dear Seq Fans,
Goerg Fischer has been doing some very interesting work, discovering
new connections between sequences in the OEIS.

There are two questions that have arisen recently that need answers:

Q1: these two sequences,
A045318 Primes p such that x^8 = 3 has no solution mod p.
A301916 Primes which divide numbers of the form 3^k + 1.
are closely related. In fact they are almost the same, the terms
in the latter but not in the former being

 2, 769, 1297, 6529, 7057, 8017, 8737, 12097, 12289. ...

for which there is now a new entry, A320481.

The question is, what are these primes?

Q2: take a look at A027595 and A007212
Are they the same?
A027595 is defined as follows:
Sequence satisfies T^2(a)=a, where T is defined below.
Define T:a->b by: given a1<=a2<=..., let b(n) = number of ways of
partitioning n into parts from a1, a2, ... such that parts = 0 mod 5 do not
occur more than once.

A007212 is paired with A007213, and the definition seems to say that they
get swapped under the "partition" transform and its inverse.  (They are
from work I did with Mira Bernstein 25 years ago.)

The partition transform is defined by (in Maple):


# Procedure PARTITION takes a list l,
# and determines the number P(i) of ways of
# partitioning an integer i between 1 and max(list) into numbers from the
list
# (NOT counting repetitions). The output is the list [P(1), P(2),...].
# The list l (sorry about using that unreadable letter) should start with
1, otherwise the 0's in the output will be skipped.
# A second argument can be used to cut down the length of the output, which
can
# otherwise quickly grow unmanageable under iterations: the argument is a
# number >=0, which specifies how much longer the output list should be than
# the input list.

PARTITION := proc()
        local f, g, i,translist,n,lp,l:
        l:=args[1]:
        if whattype(l) <> list then RETURN([]); fi:
        l:=convert(l,set);
        lp:=convert(l,list);
        if nargs=1 then n:=max(op(lp)) else n:=args[2] fi:
        f := 1:
        for i to nops(lp) while lp[i]<=n do
                f:=f/(1-t^op(i,lp)) od:
        g:=taylor(f, t=0, n+1):
        translist := []:
        for i from 2 to nops(g)/2-1 do
                translist:=[op(translist),op(2*i-1,g)]:
        od:
        translist;
end:

and  has inverse transform:

PARTITIONi:=proc(l)
        local t,i,j,l1:
if whattype(l) <> list then RETURN([]); fi:
        t:=EULERi(l);
        l1:=[]:
        for i to nops(t) do
                for j from 1 to t[i] do l1:=[op(l1),i] od:
        od:
l1;
end:

where

EULERi:=proc(b)  local a,c,i,d:
if whattype(b) <> list then RETURN([]); fi:
c:=[]:
for i to nops(b) do c:=[op(c),i*b[i]-add(c[d]*b[i-d], d=1..i-1)]: od:
a:=[]:
for i to nops(b) do a:=[op(a),(1/i)*add(mob(i,d)*c[d] , d=1..i)]: od:
RETURN(a);
end:

(These are from the "Transforms" link at the foot of every OEIS page)

If it wasn't for the strange (Rogers-Ramanujan-like?) condition that parts
= 0 mod 5 do not occur more than once, Q2 would sound pretty
straightforward.  In both cases there is a pair of sequences that get
exchanged, and in both cases the generating functions are partition-like.

But I haven't had time to study either question.  Over to you Seq Fans!