A259385 Palindromic numbers in bases 2 and 9 written in base 10.

0, 1, 3, 5, 7, 127, 255, 273, 455, 6643, 17057, 19433, 19929, 42405, 1245161, 1405397, 1786971, 2122113, 3519339, 4210945, 67472641, 90352181, 133638015, 134978817, 271114881, 6080408749, 11022828069, 24523959661, 25636651261, 25726334461, 28829406059, 1030890430479, 1032991588623, 1085079274815, 1616662113341

Offset: 1

Author

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

Keywords

• nonn
• base

Examples

			273 is in the sequence because 273_10 = 333_9 = 100010001_2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..44
• A.H.M. Smeets, Scatterplot of log_2(number is palindromic in base 2 and base b) versus b, for b in {3,5,6,7,9,10}
• Index entries for sequences related to palindromes

Crossrefs

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376, A097930, A182234, A259377, A259378, A249156, A097931, A259380, A259381, A259382, A259383, A259384, A099145, A259385, A259386, A259387, A259388, A259389, A259390, A099146, A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250409, A250410, A250411, A099165, A250412.

Programs

Mathematica

(* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 9]; If[palQ[pp, 2], AppendTo[lst, pp]; Print[pp]]; k++]; lst

Python

def nextpal(n, base): # m is the first palindrome successor of n in base base
    m, pl = n+1, 0
    while m > 0:
        m, pl = m//base, pl+1
    if n+1 == base**pl:
        pl = pl+1
    n = n//(base**(pl//2))+1
    m, n = n, n//(base**(pl%2))
    while n > 0:
        m, n = m*base+n%base, n//base
    return m
n, a2, a9 = 0, 0, 0
while n <= 30:
    if a2 < a9:
        a2 = nextpal(a2,2)
    elif a9 < a2:
        a9 = nextpal(a9, 9)
    else: # a2 == a9
        print(a2, end=",")
        a2, a9, n = nextpal(a2,2), nextpal(a9,9), n+1 # A.H.M. Smeets, Jun 03 2019

Formula

Intersection of A006995 and A029955.