A367871 a(n) is the least prime q such that A367798(n)^2 is the sum of q and its reversal.

2, 29, 20147, 261104399, 20870609999, 2059108419947, 2080783998959, 200000140570007, 206841324099959, 261359249999999, 20401390509044927, 20421109564999967, 20000105691609287, 27180442947919997, 20105039549690939, 22040085159209699, 24000605788991999, 2008220921060899607, 2008804724799599927

Offset: 1

Author

Robert Israel, Dec 03 2023

Keywords

• nonn
• base

Comments

a(n) is the first term q of A367796 such that A056964(q) = A367798(n)^2.

Examples

			a(4) = 261104399 because A367798(3) = 35419 and 35419^2 = 1254505561 = 261104399 + 993401162 and 261104399 is the first prime that works.
		

Crossrefs

Cf. A056964, A367796, A367798, A367900.

Programs

Maple

f:= proc(n) local y, c, d, dp, i, delta, m;
 y:= convert(n^2, base, 10);
 d:= nops(y);
 if d::even then
    if y[-1] <> 1 then return false fi;
    dp:= d-1;
    y:= y[1..-2];
    c[dp]:= 1;
 else
    dp:= d;
    c[dp]:= 0;
 fi;
 c[0]:= 0;
 for i from 1 to floor(dp/2) do
    delta:= y[i] - y[dp+1-i] - c[i-1] - 10*c[dp+1-i];
    if delta = 0 then c[dp-i]:= 0; c[i]:= 0;
    elif delta = -1 then c[dp-i]:= 1; c[i]:= 0;
    elif delta = -10 then c[dp-i]:= 0 ; c[i]:= 1;
    elif delta = -11 then c[dp-i]:= 1; c[i]:= 1;
    else return false
    fi;
    if y[i] + 10*c[i] - c[i-1] < 0  or (i=1 and y[i]+10*c[i]-c[i-1]=1) then return false fi;
  od;
  m:= (dp+1)/2;
  delta:= y[m] + 10*c[m] - c[m-1];
  if not member(delta, [seq(i, i=0..18, 2)]) then return false fi;
  [seq(y[i]+ 10*c[i]-c[i-1], i=1..m)]
end proc:
g:= proc(L) local T, d, t, p, x, i; uses combinat;
  d:= nops(L);
  T:= cartprod([select(t -> t[1]::odd, [seq([L[1]-x, x], x=max(1, L[1]-9)..min(L[1], 9))]),
    seq([seq([L[i]-x, x], x=max(0, L[i]-9)..min(9, L[i]))], i=2..d-1)]);
  while not T[finished] do
    t:= T[nextvalue]();
    p:= add(t[i][1]*10^(i-1), i=1..d-1) + L[-1]/2 * 10^(d-1) +
      add(t[i][2]*10^(2*d-i-1), i=1..d-1);
    if isprime(p) then return p fi;
  od;
-1
end proc:
p:= 11: Q:=29:
while p < 10^8 do
  p:= nextprime(p);
  d:= 1+ilog10(p^2);
  if d::even and p^2 >= 2*10^(d-1) then p:= nextprime(floor(10^(d/2)));  fi;
  v:= f(p);
  if v = false then next fi;
  q:= g(v);
  if q = -1 then next fi;
  Q:= Q, q;
od:
Q;

Formula

A056964(a(n)) = A367798(n)^2.