A260726 a(n) = smallest palindrome k > n such that k/n is a square; a(n) = 0 if no solution exists.

4, 8, 363, 484, 5445, 46464, 252, 2138312, 12321, 0, 44, 23232, 31213, 686, 53187678135, 44944, 272, 24642, 171, 0, 525, 88, 575, 46464, 5221225, 62426, 36963, 252, 464, 0, 1783799973871, 291080192, 2112, 4114, 53235, 69696, 333, 20102, 93639, 0, 656, 858858

Offset: 1

Author

Pieter Post, Jul 30 2015

Keywords

• nonn
• base

Comments

If n is a multiple of 10 then a(n) = 0, since no palindrome ends in 0.

Up to 200 only 3 terms are currently unknown, a(125) > 5.2*10^28, a(177) > 3.5*10^27 and a(185) > 4.5*10^27. See Links for a table of known values. - Giovanni Resta, Aug 05 2015

If a(125) > 0, then the first 3 digits of a(125) are 521 and the last 3 digits of a(125) are 125. Proof: Let m^2 = a(125)/125. Then m is odd as otherwise 125*m^2 is a multiple of 10 which is not a palindrome. Since m is odd, m^2 == 1 mod 8 and thus 125*m^2 == 125 mod 1000. - Chai Wah Wu, Mar 31 2016

Examples

			a(3) = 363, because 363/3 = 11^2. 363 * 3 = 1089, which is also a square.
a(15) = 53187678135, because 53187678135/15 = 59547^2 and 53187678135 * 15 = 893205^2.
		

Links

• Giovanni Resta, Known values up to a(200)

Crossrefs

Cf. A002113, A000290, A023108, A061563.

Programs

Maple

ispali:= proc(n) local L; L:= convert(n,base,10); ListTools:-Reverse(L)=L end proc:
f:= proc(n) local m;
   if n mod 10 = 0 then return 0 fi;
   for m from 2 to 10^6 do if ispali(m^2*n) then return m^2*n fi od:
   -1  # signals time-out
end proc:
seq(f(n), n=1..50); # Robert Israel, Aug 21 2015

Mathematica

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; a[n_] := If[ Mod[n, 10] == 0, 0, Block[{q = 2}, While[! palQ[q^2 * n], q++]; q^2 * n]]; Array[a, 42] (* Giovanni Resta, Aug 18 2015 *)

Python

def a(n):
    if n % 10 == 0: return 0
    for c in range(2, 10**8):
        k = str(n * c**2)
        if k == k[::-1]:
            return int(k)
    return -1
print(*[a(n) for n in range(1, 43)], sep=', ')
# Corrected by David Radcliffe, May 10 2025

Extensions

Missing a(13) from Giovanni Resta, Aug 05 2015