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A260726 a(n) = smallest palindrome k > n such that k/n is a square; a(n) = 0 if no solution exists. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..42.

Giovanni Resta, Known values up to a(200)

EXAMPLE

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.

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 *)

PROG

(Python)

for k in range(1, 150):

....c=2

....while c < 10**8:

........kk=k*c**2

........if kk==int(str(kk)[::-1])

............print (k, kk)

............c=10**9

........c=c+1

CROSSREFS

Cf. A002113, A000290, A023108, A061563.

Sequence in context: A065022 A012994 A136386 * A308553 A177860 A013443

Adjacent sequences:  A260723 A260724 A260725 * A260727 A260728 A260729

KEYWORD

nonn,base

AUTHOR

Pieter Post, Jul 30 2015

EXTENSIONS

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

STATUS

approved

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Last modified September 16 06:21 EDT 2019. Contains 327090 sequences. (Running on oeis4.)