A367798 Primes p such that p^2 is the sum of a prime and its reverse.

2, 11, 307, 35419, 347651, 3091643, 3417569, 30001253, 34158919, 35515619, 305524927, 312123463, 313513517, 327371987, 337660679, 348898811, 352023571, 3013005397, 3051026827, 3147298717, 3149171717, 3171167353, 3175236553, 3226951193, 3248169343, 3306563683, 3350101739, 3366748421, 3403341569

Offset

1,1

Comments

Primes p such that p^2 = A056964(q) for some term q of A367796.

Do all terms except for 2 and 11 start with 3?

From Ivan N. Ianakiev, Dec 16 2023: (Start)

To prove that for all n > 2 the first digit of a(n) is 3 is easy if the number of digits of q is odd. Sketch of a proof: Let p^2 = q + rev(q). We observe that:

a) the last digit of q must be 1, 3, 7, or 9;

b) the last digit of rev(q) cannot be zero, since the first digit of q cannot be zero;

c) the last digit of rev(q) cannot be odd, since the last digit of p^2 cannot be even (if it were, that would imply that p is even).

The rest is just a matter of bookkeeping.

To prove that for n > 2 the number of digits of q cannot be even is probably much more difficult. (End)

Links

Table of n, a(n) for n=1..29.

Example

a(1) = 2 is a term because 2^2 = 4 = 2 + 2 with 2 prime.
a(2) = 11 is a term because 11^2 = 121 = 29 + 92 with 11 and 29 prime.
a(3) = 307 is a term because 307^2 = 94249 = 20147 + 74102 with 307 and 20147 prime.
a(4) = 35419 is a term because 35419^2 = 1254505561 = 261104399 + 993401162 with 35419 and 261104399 prime.
a(5) = 347651 is a term because 347651^2 = 120861217801 = 20870609999 + 99990607802 with 347651 and 20870609999 prime.
a(6) = 3091643 is a term because 3091643^2 = 9558256439449 = 2059108419947 + 7499148019502 with 3091643 and 2059108419947 prime.
a(7) = 3417569 is a term because 3417569^2 = 11679777869761 = 2080783998959 + 9598993870802 with 3417569 and 2080783998959 prime.
a(8) = 30001253 is a term because 30001253^2 = 900075181570009 = 200000140570007 + 700075041000002 with 30001253 and 200000140570007 prime.
a(9) = 34158919 is a term because 34158919^2 = 1166831747248561 = 206841324099959 + 959990423148602 with 34158919 and 206841324099959 prime.
a(10) = 35515619 is a term because 35515619^2 = 1261359192953161 = 261359249999999 + 999999942953162 with 35515619 and 261359249999999 prime.

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: R:= 2, 11:
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;
  R:= R, p;
od:
R;

Crossrefs

Cf. A056964, A367796, A367900, A367871.

Sequence in context: A350932 A369947 A198894 * A222206 A229268 A309068

Adjacent sequences: A367795 A367796 A367797 * A367799 A367800 A367801

Keyword

nonn,base

Author

Robert Israel, Nov 30 2023

Status

approved