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

1,3

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)

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)

Example

89478485 = 101010101010101010101010101_2 = 11111111111111_4 = 525252525_8.

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 *)

Prog

(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()]
(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
(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
(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

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.

Sequence in context: A273254 A222644 A181904 * A318546 A076513 A222611

Adjacent sequences: A319581 A319582 A319583 * A319585 A319586 A319587

Keyword

nonn,base

Author

Jeremias M. Gomes, Sep 23 2018

Status

approved