OFFSET
1,1
COMMENTS
For given n, it seems there is an infinity of pairs (p,q) = (p0,q0), (p1, q1), (p2, q2), ... where p is the smallest p and q the smallest q: p=p0=min(p0, p1, p2, ...) and q = q0=min(q0, q1, ...).
Conjecture: Given an integer n, there always exists a pair (p, q) such that f(p) = f(n) + q*f(n+1).
Consequence: if the conjecture is true, then the set of prime numbers of the form k^2+1 is infinite because, by induction, there exists a pair (p', q') such that f(p') = f(p-1) + q'*f(p), f(p') > f(p).
LINKS
Michel Lagneau, 90 first pairs (p,q)
EXAMPLE
The pair (11, 76) is in the sequence because A002496(11) = A002496(2) + 76*A002496(3) and 1297=5+76*17.
+----+------+-----+------+---------------------------------------------+
| n | f(n) | p | q | f(p)=f(n)+q*f(n+1) |
+----+------+-----+------+----------------------+----------------------+
| 1 | 2 | 3 | 3 | f(3)=f(1)+3*f(2) | 17=2+3*5 |
| 2 | 5 | 11 | 76 | f(11)=f(2)+76*f(3) | 1297=5+76*17 |
| 3 | 17 | 49 | 2432 | f(49)=f(3)+2432*f(4) | 90001=17+2432*37 |
| 4 | 37 | 113 | 9980 | f(113)=f(4)+9980*f(5)| 1008017=37+9980*101 |
| 5 | 101 | 55 | 748 | f(55)=f(5)+748*f(6) | 147457=101+748*197 |
| 6 | 197 | 166 | 9420 | f(166)=f(6)+9420*f(7)| 2421137=197+9420*257 |
MAPLE
T:=array(1..30000):k:=0:
nn:=500000:
for m from 1 to nn do:
if isprime(m^2+1)
then
k:=k+1:T[k]:=m^2+1:
else
fi:
od:
for n from 1 to 32 do:
ii:=0:r:=T[n]:q:=T[n+1]:
for i from 1 to k while(ii=0) do:
p:=T[i]:r1:=irem(p, q):
if r1=r and p>q
then
ii:=1:x:=(T[i]-T[n])/T[n+1]:printf(`%d, `, i):
printf(`%d, `, x):
else
fi:
od:
od:
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Michel Lagneau, Mar 21 2022
STATUS
approved