A016041 -id:A016041 - cp's OEIS frontend

A095741 Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].

2, 2, 3, 3, 7, 12, 23, 40, 94, 142, 271, 480, 856, 1721, 3099, 5572, 10799, 20782, 39468, 72672, 139867, 274480, 520376, 986318, 1914097, 3726617, 7107443, 13682325, 26430797, 51412565, 99204128, 190457946, 372035117, 727434192, 1407026351, 2724590109, 5315491839

Offset: 1

Antti Karttunen, Jun 12 2004

Note that there are no such primes in any range ]2^(2n-1),2^2n], as all even-length binary palindromes are divisible by 3 (cf. A048702).

The ratio a(n)/A036378(2n) converges as follows: 1, 0.4, 0.230769, 0.069767, 0.051095, 0.025862, 0.014268, 0.007006, 0.00461, 0.00193, 0.00101, 0.000487, 0.000235, 0.000127, 0.000061, 0.000029

Bisection of the first diagonal of triangle A095759.

Cf. A095731, A117773.

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Table[Length[Select[Range[2^(2n), 2^(2n + 1)], palindromicQ[#, 2] && PrimeQ[#] &]], {n, 10}] (* Alonso del Arte, Jan 13 2012 *)

m=vector(65536);u=vector(#m);u[1]=1;for(b=1,#m-1,c=b;e=2^floor(log(b+.5)/log(2));d=0;u[b+1]=e;while(c>0,d=d+e*(c%2);c=floor(c/2);e=e/2);m[b+1]=d);for(x=0,31,h=0;y=2^x;for(w=y,2*y-1,if(x<16,v1=4*y*w+m[w+1];v2=v1+2*y,w1=floor(w/65536);w2=w-65536*w1;v1=262144*y*w1+4*y*w2+65536*u[w1+1]/u[w2+1]*m[w2+1]+m[w1+1];v2=v1+2*y);if(isprime(v1),h++);if(isprime(v2),h++));print(2*x+3" bits: "h)) \\ Martin Raab, Jan 13 2012

a(n) = A117773(2*n+1). - Chai Wah Wu, Jul 05 2019

a(27)-a(32) from Martin Raab, Oct 20 2015
a(33)-a(35) from Chai Wah Wu, Jul 05 2019
a(36)-a(37) from Chai Wah Wu, Jul 11 2019

A194097 Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).

Original entry on oeis.org

8, 1, 5, 7, 1, 0, 1, 9, 6, 2, 9, 0

Offset: 0

Kausthub Gudipati, Aug 15 2011

			Partial sum is 0.81571010760865175... if primes up to 40 binary digits are included; partial sum is 0.8157101534554485... if summed to 42 binary digits. - _R. J. Mathar_, Aug 27 2011
Partial sum is 0.81571019628323164... for primes up to 66 bits. - _Martin Raab_, Jan 13 2012

Ranta, Prime Curios!

Cf. A095741, A118064, A244162.

a(6)-a(11) from Martin Raab, Jan 13 2012

Edited by Jon E. Schoenfield, Nov 27 2016

A164124 First differences of A016041.

Original entry on oeis.org

0, 2, 2, 10, 14, 42, 34, 20, 130, 56, 130, 750, 260, 118, 48, 168, 44, 48, 3010, 224, 76, 368, 312, 112, 230, 616, 780, 152, 160, 272, 9186, 720, 192, 1144, 176, 192, 1356, 1384, 128, 192, 720, 448, 1718, 192, 1240, 624, 320, 96, 588, 864, 720, 792, 544

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

Differences between successive primes that are palindromic in base 2.

3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879, ...

Cf. A029971, A016041, A164125. - Vladimir Joseph Stephan Orlovsky, Aug 10 2009

f[n_]:=FromDigits[RealDigits[n,2][[1]]]==FromDigits[Reverse[RealDigits[n,2][[1]]]]; a=3;lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p-a];a=p],{n,2,8!,1}];lst
Join[{0},Differences[Select[Prime[Range[5000]],IntegerDigits[#,2]== Reverse[ IntegerDigits[ #,2]]&]]] (* Harvey P. Dale, Aug 17 2024 *)

Definition corrected by Charles R Greathouse IV, Oct 08 2009

A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, 18481, 19391, 19891, 19991

Offset: 1

N. J. A. Sloane, Simon Plouffe

Every palindrome with an even number of digits is divisible by 11, so 11 is the only member of the sequence with an even number of digits. - David Wasserman, Sep 09 2004
This holds in any number base A006093(n), n>1. - Lekraj Beedassy, Mar 07 2005 and Dec 06 2009
The log-log plot shows the fairly regular structure of these numbers. - T. D. Noe, Jul 09 2013
Conjecture: The only primes with palindromic prime indices that are palindromic primes themselves are 3, 5 and 11. Tested for the primes with the first 8000000 palindromic prime indices. - Ivan N. Ianakiev, Oct 10 2014
It follows from the above conjecture that 2 is the only k such that k, prime(k), prime(m) = k + prime(k) and m are all palindromic primes. - Ivan N. Ianakiev, Mar 17 2025
Banks, Hart, and Sakata derive a nontrivial upper bound for the number of prime palindromes n <= x as x -> oo. It follows that almost all palindromes are composite. The results hold in any base. The authors use Weil's bound for Kloosterman sums. - Jonathan Sondow, Jan 02 2018
Number of terms < 100^k, k >= 1: 5, 20, 113, 781, 5953, 47995, 401698, .... - Robert G. Wilson v, Jan 03 2018, corrected by M. F. Hasler, Dec 19 2024
Initially the above comment listed 4, 20, 113, ... which is the number of terms less than 10, 1000, 10^5, ..., i.e., up to 10^(2k-1), k >= 1. The number of terms < 10^k are the cumulative sums of A016115(n) (number of prime palindromes with n digits) up to n = k. - M. F. Hasler, Dec 19 2024

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 120-121.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..47995 (all palindromic primes with fewer than 12 digits), Oct 14 2015, extending earlier b-files from T. D. Noe and A. Olah.
W. D. Banks, D. N. Hart, and M. Sakata, Almost all palindromes are composite, Math. Res. Lett., 11 No. 5-6 (2004), 853-868.
K. S. Brown, On General Palindromic Numbers
C. K. Caldwell, "Top Twenty" page, Palindrome
Patrick De Geest, World!Of Palindromic Primes
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 10, 18.
T. D. Noe, Log-log plot of the first 401696 terms
I. Peterson, Math Trek, Palindromic Primes
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
M. Shafer, First 401066 Palprimes
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Palindromic Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Palindromic prime
Index entries for sequences related to palindromes

A007500 = this sequence union A006567.
Subsequence of A188650; A188649(a(n)) = a(n); see A033620 for multiplicative closure. [Reinhard Zumkeller, Apr 11 2011]
Cf. A016041, A029732, A069469, A117697, A046942, A032350 (Palindromic nonprime numbers).
Cf. A016115 (number of prime palindromes with n digits).

Filtered([1..20000],n->IsPrime(n) and ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019

a002385 n = a002385_list !! (n-1)
a002385_list = filter ((== 1) . a136522) a000040_list
-- Reinhard Zumkeller, Apr 11 2011

ff := proc(n) local i,j,k,s,aa,nn,bb,flag; s := n; aa := convert(s,string); nn := length(aa); bb := ``; for i from nn by -1 to 1 do bb := cat(bb,substring(aa,i..i)); od; flag := 0; for j from 1 to nn do if substring(aa,j..j)<>substring(bb,j..j) then flag := 1 fi; od; RETURN(flag); end; gg := proc(i) if ff(ithprime(i)) = 0 then RETURN(ithprime(i)) fi end;
rev:=proc(n) local nn, nnn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n) if n=rev(n) and isprime(n)=true then n else fi end: seq(a(n),n=1..20000); # rev is a Maple program to revert a number - Emeric Deutsch, Mar 25 2007
# A002385 Gets all base-10 palindromic primes with exactly d digits, in the list "Res"
d:=7; # (say)
if d=1 then Res:= [2,3,5,7]:
elif d=2 then Res:= [11]:
elif d::even then
    Res:=[]:
else
    m:= (d-1)/2:
    Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
    Res:=[]: for x in Res2 do if isprime(x) then Res:=[op(Res),x]; fi: od:
fi:
Res; # N. J. A. Sloane, Oct 18 2015

Select[ Prime[ Range[2100] ], IntegerDigits[#] == Reverse[ IntegerDigits[#] ] & ]
lst = {}; e = 3; Do[p = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[p], AppendTo[lst, p]], {n, 10^e - 1}]; Insert[lst, 11, 5] (* Arkadiusz Wesolowski, May 04 2012 *)
Join[{2,3,5,7,11},Flatten[Table[Select[Prime[Range[PrimePi[ 10^(2n)]+1, PrimePi[ 10^(2n+1)]]],# == IntegerReverse[#]&],{n,3}]]] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Apr 22 2016 *)
genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 1; lst = {2, 3, 5, 7}; While[k < 19, p = Select[genPal[k], PrimeQ];
If[p != {}, AppendTo[lst, p]]; k++]; Flatten@ lst (* RGWv *)
Select[ Prime[ Range[2100]], PalindromeQ] (* Jean-François Alcover, Feb 17 2018 *)
NestList[NestWhile[NextPrime, #, ! PalindromeQ[#2] &, 2] &, 2, 41] (* Jan Mangaldan, Jul 01 2020 *)

is(n)=n==eval(concat(Vecrev(Str(n))))&&isprime(n) \\ Charles R Greathouse IV, Nov 20 2012

forprime(p=2,10^5, my(d=digits(p,10)); if(d==Vecrev(d),print1(p,", "))); \\ Joerg Arndt, Aug 17 2014

A002385_row(n)=select(is_A002113, primes([10^(n-1),10^n])) \\ Terms with n digits. For larger n, better filter primes in palindromes. - M. F. Hasler, Dec 19 2024

from itertools import chain
from sympy import isprime
A002385 = sorted((n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**5)),(int(str(x)+str(x)[-2::-1]) for x in range(1,10**5))) if isprime(n))) # Chai Wah Wu, Aug 16 2014

from sympy import isprime
A002385 = [*filter(isprime, (int(str(x) + str(x)[-2::-1]) for x in range(10**5)))]
A002385.insert(4, 11)  # Yunhan Shi, Mar 03 2023

from sympy import isprime
from itertools import count, islice, product
def A002385gen(): # generator of palprimes
    yield from [2, 3, 5, 7, 11]
    for d in count(3, 2):
        for last in "1379":
            for p in product("0123456789", repeat=d//2-1):
                left = "".join(p)
                for mid in [[""], "0123456789"][d&1]:
                    t = int(last + left + mid + left[::-1] + last)
                    if isprime(t):
                        yield t
print(list(islice(A002385gen(), 46))) # Michael S. Branicky, Apr 13 2025

[n for n in (2..18181) if is_prime(n) and Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018

Intersection of A000040 (primes) and A002113 (palindromes).
A010051(a(n)) * A136522(a(n)) = 1. [Reinhard Zumkeller, Apr 11 2011]
Complement of A032350 in A002113. - Jonathan Sondow, Jan 02 2018

More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000

Comment from A006093 moved here by Franklin T. Adams-Watters, Dec 03 2009

A006995 Binary palindromes: numbers whose binary expansion is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 73, 85, 93, 99, 107, 119, 127, 129, 153, 165, 189, 195, 219, 231, 255, 257, 273, 297, 313, 325, 341, 365, 381, 387, 403, 427, 443, 455, 471, 495, 511, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein, L. J. Upton

If b > 1 is a binary palindrome then both (2^(m+1) + 1)*b and 2^(m+1) + 2^m - b are also, where m = floor(log_2(b)). - Hieronymus Fischer, Feb 18 2012
Floor and ceiling: If d > 0 is any natural number, then A206913(d) is the greatest binary palindrome <= d and A206914(d) is the least binary palindrome >= d. - Hieronymus Fischer, Feb 18 2012
The greatest binary palindrome <= the n-th non-binary-palindrome is that binary palindrome with number A154809(n)-n+1. The corresponding formula identity is: A206913(A154809(n)) = A006995(A154809(n)-n+1). - Hieronymus Fischer, Mar 18 2012
From Hieronymus Fischer, Jan 23 2013: (Start)
The number of binary digits of a(n) is A070939(a(n)) = 1 + floor(log_2(n)) + floor(log_2(n/3)), for n > 1.
Also: A070939(a(n)) = A070939(n) + A070939(floor(n/3)) - 1, for n <> 2. (End)
Rajasekaran, Shallit, & Smith show that this is an additive basis of order 4. - Charles R Greathouse IV, Nov 06 2018

			a(3) = 3, since 3 = 11_2 is the 3rd symmetric binary number;
a(6) = 9, since 9 = 1001_2 is the 6th symmetric binary number.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
William D. Banks and Igor E. Shparlinski, Average Value of the Euler Function on Binary Palindromes, Bulletin Polish Acad. Sci. Math., Vol. 54 (2006), pp. 95-101, alternative link.
Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatshefte für Mathematik, Vol. 161, No. 1 (2010), pp. 85-114, alternative link. doi:10.1007/s00605-009-0126-y.
Kritkhajohn Onphaeng, Tammatada Khemaratchatakumthorn, Phakhinkon Napp Phunphayap, and Prapanpong Pongsriiam, Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions, Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8. See p. 2.
Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.
Aayush Rajasekaran, Jeffrey Shallit and Tim Smith, Sums of Palindromes: an Approach via Nested-Word Automata, preprint arXiv:1706.10206 [cs.FL], Jun 30 2017.
Aayush Rajasekaran, Jeffrey Shallit and Tim Smith, Additive Number Theory via Automata Theory, Theory of Computing Systems, Vol. 64 (2020), pp. 542-567.
L. J. Upton, Letter to N. J. A. Sloane with attachments, Oct. 1993.
Index entries for sequences that are an additive basis, order 4.
Index entries for sequences related to binary expansion of n
Index entries for sequences related to palindromes

See A057148 for the binary representations.
Cf. A178225, A005408, A164126, A154809 (complement).
Cf. A002113, A048700, A048701, A206913, A206914, A244162.
Even numbers that are not the sum of two terms: A241491, A261678, A262556.
Cf. A145799.
Primes: A016041 and A117697.
Cf. A000051 (a subsequence).

a006995 n = a006995_list !! (n-1)
a006995_list = 0 : filter ((== 1) . a178225) a005408_list
-- Reinhard Zumkeller, Oct 21 2011

[n: n in [0..850] | Intseq(n,2) eq Reverse(Intseq(n,2))];  // Bruno Berselli, Aug 29 2011

dmax:= 15; # to get all terms with at most dmax binary digits
revdigs:= proc(n)
  local L, Ln, i;
  L:= convert(n,base,2);
  Ln:= nops(L);
  add(L[i]*2^(Ln-i),i=1..Ln);
end proc;
A:= {0,1}:
for d from 2 to dmax do
  if d::even then
    A:= A union {seq(2^(d/2)*x + revdigs(x),x=2^(d/2-1)..2^(d/2)-1)}
  else
    m:= (d-1)/2;
    B:={seq(2^(m+1)*x + revdigs(x),x=2^(m-1)..2^m-1)};
    A:= A union B union map(`+`,B,2^m)
  fi
od:
A;  # Robert Israel, Aug 17 2014

palQ[n_Integer, base_Integer] := Module[{idn=IntegerDigits[n, base]}, idn==Reverse[idn]]; Select[Range[1000], palQ[ #, 2]&]
Select[ Range[0, 1000], # == IntegerReverse[#, 2] &] (* Robert G. Wilson v, Feb 24 2018 *)
Select[Range[0, 1000], PalindromeQ[IntegerDigits[#, 2]]&] (* Jean-François Alcover, Mar 01 2018 *)

for(n=0,999,n-subst(Polrev(binary(n)),x,2)||print1(n,",")) \\ Thomas Buchholz, Aug 16 2014

for(n=0,10^3, my(d=digits(n,2)); if(d==Vecrev(d), print1(n,", "))); \\ Joerg Arndt, Aug 17 2014

is_A006995(n)=Vecrev(n=binary(n))==n \\ M. F. Hasler, Feb 23 2018

A006995(n,m=logint(n,2),c=1<<(m-1),a,d)={if(n>=3*c,a=n-3*c;d=2*c^2,a=n-2*c;n%2*c+d=c^2)+sum(k=1,m-2^(n<3*c),if(bittest(a,m-1-k),1<>k))+(n>2)} \\ Based on Fischer's smalltalk program. - M. F. Hasler, Feb 23 2018

from itertools import count, islice, product
def bin_pals(): # generator of binary palindromes in base 10
    yield from [0, 1]
    digits, midrange = 2, [[""], ["0", "1"]]
    for digits in count(2):
        for p in product("01", repeat=digits//2-1):
            left = "1"+"".join(p)
            for middle in midrange[digits%2]:
                yield int(left + middle + left[::-1], 2)
print(list(islice(bin_pals(), 58))) # Michael S. Branicky, Jan 09 2023

def A006995(n):
    if n == 1: return 0
    a = 1<<(l:=n.bit_length()-2)
    m = a|(n&a-1)
    return (m<Chai Wah Wu, Jun 10 2024

def palgenbase2(): # generator of palindromes in base 2
    yield 0
    x, n, n2 = 1, 1, 2
    while True:
        for y in range(n,n2):
            s = format(y,'b')
            yield int(s+s[-2::-1],2)
        for y in range(n,n2):
            s = format(y,'b')
            yield int(s+s[::-1],2)
        x += 1
        n *= 2
        n2 *= 2 # Chai Wah Wu, Jan 07 2015

[n for n in (0..843) if Word(n.digits(2)).is_palindrome()] # Peter Luschny, Sep 13 2018

A006995
"Answer the n-th binary palindrome
(nonrecursive implementation)"
| m n a b c d k2 |
n := self.
n = 1 ifTrue: [^0].
n = 2 ifTrue: [^1].
m := n integerFloorLog: 2.
c := 2 raisedToInteger: m - 1.
n >= (3 * c)
  ifTrue:
   [a := n - (3 * c).
   d := 2 * c * c.
   b := d + 1.
   k2 := 1.
   1 to: m - 1
    do:
     [:k |
     k2 := 2 * k2.
     b := b + (a * k2 // c \\ 2 * (k2 + (d // k2)))]]
  ifFalse:
   [a := n - (2 * c).
   d := c * c.
   b := d + 1 + (n \\ 2 * c).
   k2 := 1.
   1 to: m - 2
    do:
     [:k |
     k2 := 2 * k2.
     b := b + (a * k2 // c \\ 2 * (k2 + (d // k2)))]].
^b // by Hieronymus Fischer, Feb 15 2013

A178225(a(n)) = 1; union of A048700 and A048701. - Reinhard Zumkeller, Oct 21 2011
From Hieronymus Fischer, Dec 31 2008, Jan 10 2012, Feb 18 2012: (Start)
Written as a decimal, a(10^n) has 2*n digits. For n > 1, the decimal expansion of a(10^n) starts with 22..., 23... or 24...:
a(1000) = 249903,
a(10^4) = 24183069,
a(10^5) = 2258634081,
a(10^6) = 249410097687,
a(10^7) = 24350854001805,
a(10^8) = 2229543293296319,
a(10^9) = 248640535848971067,
a(10^10)= 24502928886295666773.
Inequality: (2/9)*n^2 < a(n) < (1/4)*(n+1)^2, if n > 1.
lim sup_{n -> oo} a(n)/n^2 = 1/4, lim inf_{n -> oo} a(n)/n^2 = 2/9.
For n >= 2, a(2^n-1) = 2^(2n-2) - 1; a(2^n) = 2^(2n-2) + 1;
   a(2^n+1) = 2^(2n-2) + 2^(n-1) + 1; a(2^n + 2^(n-1)) = 2^(2n-1) + 1.
Recursion for n > 2: a(n) = 2^(2k-q) + 1 + 2^p*a(m), where k = floor(log_2(n-1)), and p, q and m are determined as follows:
Case 1: If n = 2^(k+1), then p = 0, q = 0, m = 1;
Case 2: If 2^k < n < 2^k+2^(k-1), then p = k-floor(log_2(i))-1 with i = n-2^k, q = 2, m = 2^floor(log_2(i)) + i;
Case 3: If n = 2^k + 2^(k-1), then p = 0, q = 1, m = 1;
Case 4: If 2^k + 2^(k-1) < n < 2^(k+1), then p = k-floor(log_2(j))-1 with j = n-2^k-2^(k-1), q = 1, m = 2*2^floor(log_2(j))+j.
Non-recursive formula:
Let n >= 3, m = floor(log_2(n)), p = floor((3*2^(m-1)-1)/n), then
a(n) = 2^(2*m-1-p) + 1 + p*(1-(-1)^n)*2^(m-1-p) + sum_{k=1 .. m-1-p} (floor((n-(3-p)*2^(m-1))/2^(m-1-k)) mod 2)*(2^k+2^(2*m-1-p-k)). [Typo at the last exponent of the third sum term eliminated by the author, Sep 05 2018]
a(n) = 2^(2*m-2) + 1 + 2*floor((n-2^m)/2^(m-1)) + 2^(m-1)*floor((1/2)*min(n+1-2^m,2^(m-1)+1)) + 3*2^(m-1)*floor((1/2)*max(n+1-3*2^(m-1),0)) + 3*sum_{j=2 .. m-1} floor((n+2^(j-1)-2^m)/2^j)*2^(m-j). [Seems correct for n > 3. - The Editors]
Inversion formula: The index of any binary palindrome b = a(n) > 0 is n = palindromicIndex(b) = ((5-(-1)^m)/2 + Sum_{k=1..[m/2]} ([b/2^k] mod 2)/2^k)*2^[m/2], where [.] = floor(.) and m = [log_2(b)].
(End)
G.f.: g(x) = x^2 + 3x^3 + sum_{j=1..oo}( 3*2^j*(1-x^floor((j+1)/2))/(1-x)*x^((1/2)-floor((j+1)/2)) + f_j(x) - f_j(1/x))*x^(2*2^floor(j/2)+3*2^floor((j-1)/2)-(1/2)), where the f_j(x) are defined as follows:
  f_1(x) = x^(1/2), and for j > 1,
  f_j(x) = x^(1/2)*sum_{i=0..2^floor((j-1)/2)-1}((3+(1/2)*sum_{k=1..floor((j-1)/2)}(1-(-1)^floor(2i/2^k))*b(j,k))*x^i), where b(j,k) = 2^(floor((j-1)/2)-k)*((3+(-1)^j)*2^(2*k+1)+4) for k > 1, and b(j,1) = (2+(-1)^j)*2^(floor((j-1)/2)+1).  - Hieronymus Fischer, Apr 04 2012
A044051(n) = (a(n)+1)/2 for n > 0. - Reinhard Zumkeller, Apr 20 2015
A145799(a(n)) = a(n). - Reinhard Zumkeller, Sep 24 2015
Sum_{n>=2} 1/a(n) = A244162. - Amiram Eldar, Oct 17 2020

Edited and extended by Hieronymus Fischer, Feb 21 2012

Edited by M. F. Hasler, Feb 23 2018

Every palindrome with an even number of digits is divisible by 11 (in base 2) and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, namely 11_2 = 3_{10}.

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero. Obviously, all palindromic numbers are balanced; cf. A016041 for base-2 palindromic primes.

These are the primes in A256082. This is the binary variant of the decimal version A256076 suggested by Eric Angelini.

cp's OEIS Frontend

A095741 Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].

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A194097 Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).

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A164124 First differences of A016041.

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A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome.

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A006995 Binary palindromes: numbers whose binary expansion is palindromic.

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A329419 Numbers all of whose divisors are binary palindromes.

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