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A319584 Numbers that are palindromic in bases 2, 4, and 8. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Sep 25 2018 *)

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

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Last modified June 25 20:02 EDT 2022. Contains 354851 sequences. (Running on oeis4.)