cp's OEIS Frontend

A352582 Two-column array read by rows, where the n-th row is the least pair of integers (p, q) such that f(p) = f(n) + q*f(n+1) where f(n) = A002496(n) is the n-th prime of the form k^2+1.

%I A352582 #32 Mar 26 2022 22:45:16
%S A352582 3,3,11,76,49,2432,113,9980,55,748,166,9420,384,39780,130,2388,271,
%T A352582 10640,867,82592,1054,103040,548,11828,578,12332,4874,1113600,2461,
%U A352582 196380,1137,27932,2426,128944,1393,35708,16086,5861020,2052,54268,9154,1437780,7981,982208
%N A352582 Two-column array read by rows, where the n-th row is the least pair of integers (p, q) such that f(p) = f(n) + q*f(n+1) where f(n) = A002496(n) is the n-th prime of the form k^2+1.
%C A352582 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, ...).
%C A352582 Conjecture: Given an integer n, there always exists a pair (p, q) such that f(p) = f(n) + q*f(n+1).
%C A352582 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).
%H A352582 Michel Lagneau, <a href="/A352582/a352582.pdf">90 first pairs (p,q)</a>
%e A352582 The pair (11, 76) is in the sequence because A002496(11) = A002496(2) + 76*A002496(3) and 1297=5+76*17.
%e A352582 +----+------+-----+------+---------------------------------------------+
%e A352582 |  n | f(n) |   p |    q |            f(p)=f(n)+q*f(n+1)               |
%e A352582 +----+------+-----+------+----------------------+----------------------+
%e A352582 |  1 |   2  |   3 |    3 | f(3)=f(1)+3*f(2)     |      17=2+3*5        |
%e A352582 |  2 |   5  |  11 |   76 | f(11)=f(2)+76*f(3)   |    1297=5+76*17      |
%e A352582 |  3 |  17  |  49 | 2432 | f(49)=f(3)+2432*f(4) |   90001=17+2432*37   |
%e A352582 |  4 |  37  | 113 | 9980 | f(113)=f(4)+9980*f(5)| 1008017=37+9980*101  |
%e A352582 |  5 | 101  |  55 |  748 | f(55)=f(5)+748*f(6)  |  147457=101+748*197  |
%e A352582 |  6 | 197  | 166 | 9420 | f(166)=f(6)+9420*f(7)| 2421137=197+9420*257 |
%p A352582 T:=array(1..30000):k:=0:
%p A352582 nn:=500000:
%p A352582   for m from 1 to nn do:
%p A352582    if isprime(m^2+1)
%p A352582     then
%p A352582      k:=k+1:T[k]:=m^2+1:
%p A352582     else
%p A352582    fi:
%p A352582   od:
%p A352582   for n from 1 to 32 do:
%p A352582   ii:=0:r:=T[n]:q:=T[n+1]:
%p A352582    for i from 1 to k while(ii=0) do:
%p A352582      p:=T[i]:r1:=irem(p,q):
%p A352582         if r1=r and p>q
%p A352582          then
%p A352582          ii:=1:x:=(T[i]-T[n])/T[n+1]:printf(`%d, `,i):
%p A352582          printf(`%d, `,x):
%p A352582          else
%p A352582         fi:
%p A352582        od:
%p A352582       od:
%Y A352582 Cf. A002496, A348598.
%K A352582 nonn,tabf
%O A352582 1,1
%A A352582 _Michel Lagneau_, Mar 21 2022