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A083834 Palindromes of the form 5p + 1 where p is also a palindrome. Palindromes arising in A083833.

%I A083834 #21 Jun 14 2021 04:15:46
%S A083834 6,11,111,606,656,1111,11111,60106,60606,65156,65656,111111,1111111,
%T A083834 6010106,6015106,6060606,6065606,6510156,6515156,6560656,6565656,
%U A083834 11111111,111111111,601010106,601060106,601515106,601565106,606010606
%N A083834 Palindromes of the form 5p + 1 where p is also a palindrome. Palindromes arising in A083833.
%C A083834 From _Michael S. Branicky_, Jun 13 2021: (Start)
%C A083834 All terms start and end with the digit 6, except those consisting of d+1 1's, which arise from palindromes of d 2's.
%C A083834 If a(n) > 6 starts with a 6, then its second and second-to-last digits are either both 0 or both 5; and if it has 5 or more digits, its third and third-to-last digits are either both 1 or both 6; thus, terms in the latter case only start with 111, 601, 606, 651, 656.
%C A083834 Using ^ to denote repeated concatenation, contains terms of the forms 1^(d+1), arising from palindromes of the form 2^d; (60)^d 6, arising from (12)^d 1; and (65)^d 6, arising from (13)^d 1; among other patterns.
%C A083834 (Conjectures)
%C A083834 All terms contain only the digits {0, 1, 5, 6}.
%C A083834 For d odd, the only term with d digits is 1^d; equivalently, all terms starting with 6 have odd length. (End)
%H A083834 Michael S. Branicky, <a href="/A083834/b083834.txt">Table of n, a(n) for n = 1..8217</a> (all terms where p has <= 26 digits)
%t A083834 Select[5Select[Range@100000,PalindromeQ]+1,PalindromeQ] (* _Giorgos Kalogeropoulos_, Jun 11 2021 *)
%o A083834 (Python)
%o A083834 from itertools import product
%o A083834 def ispal(n): s = str(n); return s == s[::-1]
%o A083834 def pals(d, base=10): # all positive d-digit palindromes
%o A083834     digits = "".join(str(i) for i in range(base))
%o A083834     for p in product(digits, repeat=d//2):
%o A083834         if d > 1 and p[0] == "0": continue
%o A083834         left = "".join(p); right = left[::-1]
%o A083834         for mid in [[""], digits][d%2]:
%o A083834             t = int(left + mid + right)
%o A083834             if t > 0: yield t
%o A083834 def ok(pal): return ispal(5*pal+1)
%o A083834 print([5*p+1 for d in range(1, 10) for p in pals(d) if ok(p)]) # _Michael S. Branicky_, Jun 11 2021
%o A083834 (PARI) is(n) = my(x=(n-1)/5, dn=digits(n), dx); if(x!=ceil(x), return(0)); dx=digits(x); dn==Vecrev(dn) && dx==Vecrev(dx) && n>1 \\ _Felix Fröhlich_, Jun 11 2021
%Y A083834 Cf. A083829, A083830, A083832, A083833.
%K A083834 base,nonn
%O A083834 1,1
%A A083834 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003
%E A083834 Corrected and extended by _Ray Chandler_, May 21 2003