login
A272864
Palindromes that are not of the form x^2 + y^2 + z^2 where x, y, z are integers.
1
7, 55, 111, 151, 191, 252, 303, 343, 383, 444, 535, 575, 636, 727, 767, 828, 919, 959, 999, 1111, 1551, 1991, 2332, 3223, 3663, 4444, 5335, 5775, 6556, 7007, 7447, 7887, 8668, 9119, 9559, 9999, 11111, 11311, 11511, 11711, 11911, 13031, 13231, 13431, 13631, 13831, 15151, 15351, 15551, 15751
OFFSET
1,1
COMMENTS
First differences are 48, 56, 40, 40, 61, 51, 40, 40, 61, 91, 40, 61, 91, 40, 61, 91, 40, 40, 112, 440, 440, ...
This sequence is obviously infinite. (10^n - 1)/9 is a member of this sequence for all n > 2.
Proof : A004215 lists numbers of the form 4^i*(8*j+7) where i,j >= 0. So numbers of the form 8*j+7 is always in A004215 for i = 0, j >= 0. Since (10^n - 1)/9 - 7 = (10^n - 2^6)/9 = (2^n*5^n - 2^6)/9 = 2^3*(2^(n-3)*5^n - 2^3)/9, (10^n - 1)/9 - 7 is always divisible by 2^3 for n > 2, that means (10^n - 1)/9 is number of the form 8*j + 7 for n > 2. Trivially, (10^n - 1)/9 is always a palindrome for positive n, that is (10^n - 1)/9 is member of this sequence for all n > 2.
LINKS
EXAMPLE
44 is not a term because 44 = 2^2 + 2^2 + 6^2.
55 is a term because 55 = x^2 + y^2 + z^2 has no solution for integer values of x, y, z.
MAPLE
revdigs:= proc(x) local L, i;
L:= convert(x, base, 10);
add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
palis:= proc(d) local x;
if d::even then seq(x*10^(d/2)+revdigs(x), x=10^(d/2-1)..10^(d/2)-1)
else seq(x*10^((d-1)/2)+revdigs(floor(x/10)), x=10^((d-1)/2)..10^((d+1)/2)-1)
fi
end proc:
filter:= proc(x) local i;
i:= floor(padic:-ordp(x, 2)/2);
x/4^i mod 8 = 7
end proc:
select(filter, [seq(palis(d), d=1..5)]); # Robert Israel, May 09 2018
MATHEMATICA
Select[Range[16000], And[! SquaresR[3, #] > 0, Reverse@ # == # &@ IntegerDigits@ #] &] (* Michael De Vlieger, May 08 2016 *)
PROG
(PARI) isA002113(n) = {my(d = digits(n)); (subst(Polrev(d), x, 10) == n); }
isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA002113(n) && isA004215(n), print1(n, ", ")));
CROSSREFS
Sequence in context: A192718 A014637 A062212 * A121183 A291891 A217327
KEYWORD
nonn,base,look
AUTHOR
Altug Alkan, May 08 2016
STATUS
approved