A319584 Numbers that are palindromic in bases 2, 4, and 8.

0, 1, 3, 5, 63, 65, 195, 325, 341, 4095, 4097, 4161, 12291, 12483, 20485, 20805, 21525, 21845, 258111, 262143, 262145, 266305, 786435, 798915, 1310725, 1311749, 1331525, 1332549, 1376277, 1377301, 1397077, 1398101, 16515135, 16777215, 16777217, 16781313

Offset: 1

Author

Jeremias M. Gomes, Sep 23 2018

Keywords

• nonn
• base

Comments

Intersection of A006995, A014192, and A029803.

From A.H.M. Smeets, Jun 08 2019: (Start)

Intersection of A006995 and A259382.

Intersection of A014192 and A259380.

Intersection of A029803 and A097856.

All repunit numbers in base 2 with 6*k digits are included in this sequence, i.e., all terms A000225(6*k) for k >= 0.

All repunit numbers in base 4 with 2+3*k digits are included in this sequence, i.e., all terms A002450(2+3*k) for k >= 0.

All terms A000051(6*k) for k > 0 are included in this sequence.

All terms A052539(3*k) for k > 0 are included in this sequence.

In general, for sequences with palindromic numbers in the set of bases {b, b^2, ..., b^k}, gaps of size 2 occur at the term pairs (b^(k!) - 1, b^(k!) + 1). See also A319598 for b = 2 and k = 4.

The terms occur in bursts with large gaps in between as shown in the scatterplots of log_b(a(n)-a(n-1)) versus log_b(n) and log_b(1-a(n-1)/a(n)) versus log_b(n). Terms of this sequence are those with b = 2 and k = 3. For comparison, terms with b = 3 and k = 3 are also shown in these plots.

(End)

Examples

			89478485 = 101010101010101010101010101_2 = 11111111111111_4 = 525252525_8.
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..2298
• A.H.M. Smeets, Scatterplot of log_b(a(n)-a(n-1)) versus log_b(n)
• A.H.M. Smeets, Scatterplot of log_b(1-a(n-1)/a(n)) versus log_b(n)

Crossrefs

Cf. A006995 (base 2), A014192 (base 4), A029803 (base 8), A097956 (bases 2 and 4), A259380 (bases 2 and 8), A259382 (bases 4 and 8), A319598 (bases 2, 4, 8 and 16).

Cf. A000051, A000225, A002450, A052539.

Programs

Magma

[n: n in [0..2*10^7] | Intseq(n, 2) eq Reverse(Intseq(n, 2)) and Intseq(n, 4) eq Reverse(Intseq(n, 4)) and Intseq(n, 8) eq Reverse(Intseq(n, 8))]; // Vincenzo Librandi, Sep 24 2018

Mathematica

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 4] && palQ[n, 8], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)
Select[Range[0,168*10^5],AllTrue[Table[IntegerDigits[#,d],{d,{2,4,8}}],PalindromeQ]&] (* Harvey P. Dale, Jan 27 2024 *)

PARI

ispal(n, b) = my(d=digits(n, b)); Vecrev(d) == d;
isok(n) = ispal(n, 2) && ispal(n, 4) && ispal(n, 8); \\ Michel Marcus, Jun 11 2019

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 <= 100:
    if a == rev(a, 4) == rev(a, 2):
        print(a)
        n += 1
    a = nextpal(a, 8) # A.H.M. Smeets, Jun 08 2019

Sage

[n for n in (0..1000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(4)).is_palindrome() and Word(n.digits(8)).is_palindrome()]