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A256704 Palindromes of the form 4n + 1 that are divisible by 5.

%I A256704 #34 Jul 09 2018 13:03:45
%S A256704 5,505,525,545,565,585,5005,5225,5445,5665,5885,50005,50105,50205,
%T A256704 50305,50405,50505,50605,50705,50805,50905,52025,52125,52225,52325,
%U A256704 52425,52525,52625,52725,52825,52925,54045,54145,54245,54345,54445,54545,54645,54745,54845
%N A256704 Palindromes of the form 4n + 1 that are divisible by 5.
%C A256704 The sequence is infinite.
%C A256704 Creation of the Sequence. The sequence was generated based on the following known results for primitive Pythagorean triples/triangles (PPTs).
%C A256704 Let c represent the hypotenuse of a PPT and a, b be its sides. Without loss of generality, let a be the odd side and b be the even side.
%C A256704 (a) c takes the form 4n+1.
%C A256704 (b) One of a,b,c is divisible by 5.
%C A256704 Let c be divisible by 5. Thus, this sequence represents the palindromic hypotenuse of a PPT that is divisible by 5.
%C A256704 With the exception of 5, the last two digits of each term of this sequence are 05, 25, 45, 65 or 85. Thus, since the terms are palindromes, the first two digits of each term of the sequence are 50, 52, 54, 56 or 58.
%C A256704 Proof. If k is an element of the sequence, then its form must be 4n+1 for some integer n. This means that k-1=4n that is, k-1 is divisible by 4. Since an integer is divisible by 4 if and only if the number composed of its last two digits is divisible by 4, it must be that the number composed of the last two digits of k-1 is divisible by 4. With this we get the desired result.
%C A256704 Except for the terms having one or three digits, the terms of the sequence take the following forms:
%C A256704 (a) 50p05
%C A256704 (b) 52p25
%C A256704 (c) 54p45
%C A256704 (d) 56p65
%C A256704 (e) 58p85
%C A256704 where p is a palindromic number with leading zero digit(s) allowed.
%H A256704 Harvey P. Dale, <a href="/A256704/b256704.txt">Table of n, a(n) for n = 1..1000</a>
%H A256704 John Rafael M. Antalan and Richard P. Tagle, <a href="http://arxiv.org/abs/1502.06760">Numeric Palindromes in Primitive and Non-Primitive Pythagorean Triples</a>, arXiv:1502.06760 [math.NT], 2015.
%t A256704 palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[Select[4 Range[100000] + 1, palQ], Mod[#, 5] == 0 &] (* _Michael De Vlieger_, Apr 08 2015 *)
%t A256704 Select[5*Range[12000],PalindromeQ[#]&&Mod[#,4]==1&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 09 2018 *)
%o A256704 (PARI) ispal(n) = (d = digits(n)) && (Vecrev(d) == d);
%o A256704 isok(n) = !(n % 5) && !((n-1) % 4) && ispal(n); \\ _Michel Marcus_, Apr 08 2015
%K A256704 base,nonn
%O A256704 1,1
%A A256704 _John Rafael M. Antalan_, Apr 08 2015
%E A256704 More terms from _Michel Marcus_, Apr 08 2015