A083833 -id:A083833 - cp's OEIS frontend

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

6, 11, 111, 606, 656, 1111, 11111, 60106, 60606, 65156, 65656, 111111, 1111111, 6010106, 6015106, 6060606, 6065606, 6510156, 6515156, 6560656, 6565656, 11111111, 111111111, 601010106, 601060106, 601515106, 601565106, 606010606

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

From Michael S. Branicky, Jun 13 2021: (Start)
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.
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.
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.
(Conjectures)
All terms contain only the digits {0, 1, 5, 6}.
For d odd, the only term with d digits is 1^d; equivalently, all terms starting with 6 have odd length. (End)

Michael S. Branicky, Table of n, a(n) for n = 1..8217 (all terms where p has <= 26 digits)

Cf. A083829, A083830, A083832, A083833.

Select[5Select[Range@100000,PalindromeQ]+1,PalindromeQ] (* Giorgos Kalogeropoulos, Jun 11 2021 *)

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

from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all positive d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]:
            t = int(left + mid + right)
            if t > 0: yield t
def ok(pal): return ispal(5*pal+1)
print([5*p+1 for d in range(1, 10) for p in pals(d) if ok(p)]) # Michael S. Branicky, Jun 11 2021

Corrected and extended by Ray Chandler, May 21 2003

cp's OEIS Frontend

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

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