A308832 Numbers that are palindromic in bases 3, 9 and 27.

0, 1, 2, 4, 8, 10, 20, 364, 728, 730, 1460, 2920, 5840, 7300, 7381, 7462, 14600, 14681, 14762, 265720, 531440, 531442, 532171, 532900, 1062884, 1063613, 1064342, 2125768, 2128684, 2131600, 4251536, 4254452, 4257368, 5314420, 5321710, 5329000, 5373550, 5380840, 5388130, 5432680, 5439970, 5447260

Offset

1,3

Comments

This sequence is infinite because it contains terms of the forms 729^k-1 (k>=0) and 729^k+1 (k>=0).

Let L_27 be the number of digits for a(n) represented in base 27.

If L_27 is odd, the digits are either in the set {0, 1, 2} or in {0, 4, 8} or in {0, 10_10, 20_10}.

If L_27 is even, the digits are either in the set {0, 13_10} or in {0, 26_10} or in {13_10, 26_10}.

Let L_3 be the length of a(n) if represented in base 3, then L_3 in {0,1,2,3} (mod 6).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..2586

Mathematica

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]]; aQ[n_] := AllTrue[{3, 9, 27}, palQ[n, #] &]; Select[Range[0, 6*10^6], aQ] (* Amiram Eldar, Jul 04 2019 *)

Prog

(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
def rev(n, b):
    m = 0
    while n > 0:
        n, m = n//b, m*b+n%b
    return m
n, a = 1, 0
while n <= 20000:
    if a == rev(a, 9) and a == rev(a, 3):
        print(n, a)
        n = n+1
    a = nextpal(a, 27)
(PARI) nextpal(n, b) = {my(m=n+1, p = 0); while (m > 0, m = m\b; p++; ); if (n+1 == b^p, p++); n = n\(b^(p\2))+1; m = n; n = n\(b^(p%2)); while (n > 0, m = m*b + n%b; n = n\b; ); m; } \\ after Python
ispal(n, b) = my(d=digits(n, b)); Vecrev(d) == d;
lista(nn) = {my(k=0); while (k <= nn, if (ispal(k, 3) && ispal(k, 9), print1(k, ", "); ); k = nextpal(k, 27); ); } \\ Michel Marcus, Jul 04 2019

Crossrefs

Subsequence of A259386.

Cf. A319584 (palindromic in bases 2, 4 and 8).

Sequence in context: A259386 A004655 A127101 * A124849 A056654 A365002

Adjacent sequences: A308829 A308830 A308831 * A308833 A308834 A308835

Keyword

nonn,base

Author

A.H.M. Smeets, Jun 27 2019

Status

approved