A016041 -id:A016041 - cp's OEIS frontend

A265326 n-th prime minus its binary reversal.

1, 0, 0, 0, -2, 2, 0, -6, -6, 6, 0, -4, 4, -10, -14, 10, 4, 14, -30, -42, 0, -42, -18, 12, 30, 18, -12, 0, 18, 42, 0, -62, -8, -70, -20, -82, -28, -34, -62, -8, -26, 8, -62, 62, 34, -28, 8, -28, 28, 62, 82, -8, 98, 28, 0, -186, -84, -210, -60

Offset: 1

Max Barrentine, Dec 07 2015

a(n) = 0 iff A000040(n) is in A016041. - Altug Alkan, Dec 07 2015

The graph consists of a succession of parallelograms. The parallelograms end when there is a long run of mostly positive terms followed by a long run of mostly negative terms. The places where the successive parallelograms end are the primes just before a power of 2: 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, ..., which are terms with indices 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, ... (see A014234 and A007053). - N. J. A. Sloane, May 29 2016

			n=5: prime(5) = 11_10 = 1011_2, reversing gives 1101_2 = 13_10, so a(5) = 11-13 = -2.

N. J. A. Sloane, Table of n, a(n) for n = 1..23000, May 29 2016 [First 10000 terms from _Robert Israel_]
Rémy Sigrist, Colored scatterplot of the first 82025 terms (corresponding to the prime numbers < 2^20) (where the color is function of A000040(n) mod 8)
N. J. A. Sloane, Table of n, a(n) for n = 1..78498
N. J. A. Sloane and Brady Haran, Amazing Graphs, Numberphile video (2019)

Cf. A055945, A068396, A098957, A007053, A014234.

revdigs:= proc(n) local L, j;
  L:= convert(n,base,2);
  add(L[-j]*2^(j-1),j=1..nops(L))
end proc:
map(t -> t - revdigs(t),  select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Dec 08 2015

Table[# - FromDigits[Reverse@ IntegerDigits[#, 2], 2] &@ Prime@ n, {n, 60}] (* Michael De Vlieger, Dec 09 2015 *)

a098957(n) = my(v=binary(prime(n)), s); forstep(i=#v, 1, -1, s+=s+v[i]); s
a(n) = prime(n) - a098957(n); \\ Altug Alkan, Dec 07 2015

a(n) = A000040(n) - A098957(n).

a(n) = A055945(A000040(n)). - Michel Marcus, Dec 08 2015

A046485 Sum of first n palindromic primes A002385.

Original entry on oeis.org

2, 5, 10, 17, 28, 129, 260, 411, 592, 783, 1096, 1449, 1822, 2205, 2932, 3689, 4476, 5273, 6192, 7121, 17422, 27923, 38524, 49835, 61246, 73667, 86388, 99209, 112540, 126371, 140302, 154643, 169384, 184835, 200386, 216447, 232808, 249369, 266030, 283501

Offset: 1

Patrick De Geest, Sep 15 1998

The subsequence of prime partial sum of palindromic primes begins: 2, 5, 17, 5273, 7121, 154643, 283501. What is the smallest nontrivial (i.e., multidigit) palindromic prime partial sum of palindromic primes? [Jonathan Vos Post, Feb 07 2010]

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Patrick De Geest, World!Of Palindromic Primes

Cf. A002385, A007504, A046489.

Cf. A000040, A007500, A006567, A016041, A029732, A117697. [Jonathan Vos Post, Feb 07 2010]

t = {}; b = 10; Do[p = Prime[n]; i = IntegerDigits[p, b]; If[i == Reverse[i], AppendTo[t, p];(*Print[p.FromDigits[i]]*)], {n, 4000}]; Accumulate[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
Accumulate[Select[Prime[Range[10000]],IntegerDigits[#]==Reverse[ IntegerDigits[#]]&]] (* Harvey P. Dale, Aug 10 2013 *)

a(n) = Sum_{i=1..n} A002385(i) = Sum_{i=1..n} {p prime and R(p) = p, i.e., primes whose decimal expansion is a palindrome}. [Jonathan Vos Post, Feb 07 2010]

Offset set to 1 by R. J. Mathar, Feb 21 2010

A117773 Number of palindromic primes in base 2 with exactly n binary digits.

Original entry on oeis.org

0, 1, 2, 0, 2, 0, 3, 0, 3, 0, 7, 0, 12, 0, 23, 0, 40, 0, 94, 0, 142, 0, 271, 0, 480, 0, 856, 0, 1721, 0, 3099, 0, 5572, 0, 10799, 0, 20782, 0, 39468, 0, 72672, 0, 139867, 0, 274480, 0, 520376, 0, 986318, 0, 1914097, 0, 3726617, 0, 7107443, 0, 13682325, 0, 26430797, 0, 51412565, 0, 99204128, 0, 190457946, 0

Offset: 1

Martin Renner, Apr 15 2006

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

Chai Wah Wu, Table of n, a(n) for n = 1..76
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 34.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Cf. A016041, A117697, A095741.

Array[If[And[OddQ[#], # > 1], 0, Count[Prime@ Range[PrimePi[2^#] - Boole[# == 1] + 1, PrimePi[2^(# + 1) - 1]], ?(PalindromeQ[IntegerDigits[#, 2]] &)]] &, 25, 0] (* _Michael De Vlieger, Sep 29 2023 *)

a(23)-a(40) from Donovan Johnson, Dec 02 2009

a(41)-a(66) from Martin Raab, Oct 20 2015

A164126 First differences of A006995.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 4, 6, 4, 2, 12, 6, 12, 2, 8, 12, 8, 6, 8, 12, 8, 2, 24, 12, 24, 6, 24, 12, 24, 2, 16, 24, 16, 12, 16, 24, 16, 6, 16, 24, 16, 12, 16, 24, 16, 2, 48, 24, 48, 12, 48, 24, 48, 6, 48, 24, 48, 12, 48, 24, 48, 2, 32, 48, 32, 24, 32, 48, 32, 12, 32, 48, 32, 24, 32, 48, 32, 6

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

Contribution from Hieronymus Fischer, Feb 18 2012: (Start)
From the formula section it follows that a(2^m - 1 + 2^(m-1) - k) = a(2^m - 1 + k) for 0 <= k <= 2^(m-1), as well as a(2^m - 1 + 2^(m-1) - k) = 2 for k=0, 2^(m-1) and a(2^m - 1 + 2^(m-1) - k) = 6 for k=2^(m-2), hence, starting from positions n=2^m-1, the following 2^(m-1) terms form symmetric tuples limited on the left and on the right by a '2' and always having a '6' as the center element.
Example: for n = 15 = 2^4 - 1, we have the (2^3+1)-tuple (2,8,12,8,6,8,12,8,2).
Further on, since a(2^m - 1 + 2^(m-1) + k) = a(2^(m+1) - 1 - k) for 0 <= k <= 2^(m-1) an analogous statement holds true for starting positions n = 2^m + 2^(m-1) - 1.
Example: for n = 23 = 2^4 + 2^3 - 1, we find the (2^3+1)-tuple (2,24,12,24,6,24,12,24,2).
If we group the sequence terms according to the value of m=floor(log_2(n)), writing those terms together in separate lines and opening each new line for n >= 2^m + 2^(m-1), then a kind of a 'logarithmic shaped' cone end will be formed, where both the symmetry and the calculation rules become obvious. The first 63 terms are depicted below:
                          1
                          2
                          2
                        2  2
                        6  2
                      4 6  4  2
                     12 6 12  2
                8 12  8 6  8 12  8 2
               24 12 24 6 24 12 24 2
   16 24 16 12 16 24 16 6 16 24 16 12 16 24 16 2
   48 24 48 12 48 24 48 6 48 24 48 12 48 24 48 2
.
(End)
Decremented by 1, also the sequence of run lengths of 0's in A178225. - Hieronymus Fischer, Feb 19 2012

			a(1) = A006995(2) - A006995(1) = 1 - 0 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006995, A164124, A029971, A016041, A164125.

See A178225.

f[n_]:=FromDigits[RealDigits[n,2][[1]]]==FromDigits[Reverse[RealDigits[n, 2][[1]]]]; a=1;lst={};Do[If[f[n],AppendTo[lst,n-a];a=n],{n,1,8!,1}]; lst

def A164126(n):
    if n == 1: return 1
    m = (a:=1<<(l:=n.bit_length()-2))|(n&a-1)
    k = (m<Chai Wah Wu, Jun 11 2024

a(n) = A006995(n+1) - A006995(n).
Contribution from Hieronymus Fischer, Feb 17 2012: (Start)
a(4*2^m - 1) = a(6*2^m - 1) = 2;
a(5*2^m - 1) = a(7*2^m - 1) = 6 (for m > 0);
Let m = floor(log_2(n)), then
  Case 1: a(n) = 2, if n+1 = 2^(m+1) or n+1 = 3*2^(m-1);
  Case 2: a(n) = 2^(m-1), if n = 0(mod 2) and n < 3*2^(m-1);
  Case 3: a(n) = 3*2^(m-1), if n = 0(mod 2) and n >= 3*2^(m-1);
  Case 4: a(n) = 3*2^(m-1)/gcd(n+1-2^m, 2^m), otherwise.
Cases 2-4 above can be combined as
  Case 2': a(n) = (2 - (-1)^(n-(n-1)*floor(2*n/(3*2^m))))*2^(m-1)/gcd(n+1-2^m, 2^m).
Recursion formula:
Let m = floor(log_2(n)); then
  Case 1: a(n) = 2*a(n-2^(m-1)), if 2^m <= n < 2^m + 2^(m-2) - 1;
  Case 2: a(n) = 6, if n = 2^m + 2^(m-2) - 1;
  Case 3: a(n) = a(n-2^(m-2)), if 2^m + 2^(m-2) <= n < 2^m + 2^(m-1) - 1;
  Case 4: a(n) = 2, if n = 2^m + 2^(m-1) - 1;
  Case 5: a(n) = (2 + (-1)^n)*a(n-2^(m-1)), otherwise (which means 2^m + 2^(m-1) <= n < 2^(m+1)).
(End)

a(1) changed to 1 and keyword:base added by R. J. Mathar, Aug 26 2009

A262630 Base-10 representation of 1 and the primes at A262629.

Original entry on oeis.org

1, 7, 31, 127, 5113, 303049, 130677919, 8561616127, 343370835961, 398015959727917, 6536129506258687661, 136824982467292060343, 1727891550586579544797, 518772379027828374941147, 33164398702973727192477403, 91254204345537698333055497929

Offset: 1

Clark Kimberling, Oct 02 2015

			n   A262629(n)    base-10 representation
1   1                 1
2   111               7
3   11111             31
4   1111111           127
5   1001111111001     5113

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A262629. Subsequence of A016041 (except a(1)).

s = {1}; base = 2; z = 20; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#, base]], base]] &];
AppendTo[s, FromDigits[IntegerDigits[tmp, base]]], {z}]; s  (* A262629 *)
Map[FromDigits[ToString[#], base] &, s]  (* A262630 *)
(* Peter J. C. Moses, Sep 01 2015 *)

A333421 Primes that are palindromic in factorial base.

Original entry on oeis.org

3, 7, 11, 41, 127, 139, 173, 179, 191, 751, 811, 5113, 5167, 5419, 5443, 6581, 6659, 6737, 6761, 6833, 6863, 6911, 6959, 40609, 40897, 41047, 41479, 42061, 42349, 42499, 42643, 42787, 50549, 51131, 51419, 51563, 52289, 52433, 52583, 52727, 363361, 363481, 365473

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			3 is a term since it is a prime number and its factorial base representation is 11 which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Intersection of A000040 and A046807.
Cf. A007623, A108731.
Cf. A002385, A016041, A029971, A029972, A029973, A029974, A029975, A029976, A029977, A029978, A029979, A029980, A029981, A029982.

max = 9; Select[Range[0, max! - 1], PrimeQ[#] && PalindromeQ @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]] &]

A056130 Palindromic primes in bases 2 and 4.

Original entry on oeis.org

3, 5, 17, 257, 5189, 65537, 83269, 86293, 1053953, 1066049, 1134929, 1311749, 1380629, 16864513, 17060929, 17909009, 18153809, 18171217, 21251141, 22103317, 289423441, 290455889, 290735441, 336662789, 336925957, 340873541

Offset: 1

Robert G. Wilson v, Jul 29 2000

Intersection of A016041 and A029972.

Subsequence of primes of A097856. - Michel Marcus, Nov 07 2015

			5 is 101 (base 2) and 11 (base 4), that are both palindromic.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A016041 and A029972.

Do[If[PrimeQ[n], t = RealDigits[n, 4][[1]]; If[FromDigits[t] == FromDigits[Reverse[t]], s = RealDigits[n, 2][[1]]; If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]]], {n, 1, 10^8, 2}]

A164125 First differences of A029971.

Original entry on oeis.org

11, 10, 128, 22, 60, 524, 180, 156, 156, 180, 58, 180, 66, 90, 90, 66, 90, 6320, 714, 1008, 2190, 2650, 1722, 198, 1722, 41510, 810, 594, 1620, 6570, 3546, 5736, 1620, 1404, 810, 594, 1620, 522, 4428, 810, 1332, 1620, 1404, 7356, 594, 810, 3546, 2214

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

Differences between primes that are palindromic in base 3.

			a(4) = 22 = A029971(5)-A029971(4).

Cf. A164124, A029971, A016041

f[n_]:=FromDigits[RealDigits[n,3][[1]]]==FromDigits[Reverse[RealDigits[n, 3][[1]]]]; a=2;lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p-a];a=p], {n,1,8!,1}];lst

Definition simplified, initial 0 removed by R. J. Mathar, Nov 17 2009

A207039 Primes whose binary expansion is not palindromic.

Original entry on oeis.org

2, 11, 13, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317

Offset: 1

Omar E. Pol, Feb 25 2012

Intersection of A000040 and A154809.

Primes in A154809.

Cf. A000040, A006995, A016041.

Select[Table[Prime[n], {n, 200}], IntegerDigits[#, 2] != Reverse[IntegerDigits[#, 2]] &](* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A230820 Table, read by antidiagonals, of palindromic primes in base b expressed in decimal.

Original entry on oeis.org

3, 2, 5, 2, 13, 7, 2, 3, 23, 17, 2, 3, 5, 151, 31, 2, 3, 31, 17, 173, 73, 2, 3, 5, 41, 29, 233, 107, 2, 3, 5, 7, 67, 59, 757, 127, 2, 3, 5, 71, 37, 83, 257, 937, 257, 2, 3, 5, 7, 107, 43, 109, 373, 1093, 313, 2, 3, 5, 7, 73, 157, 61, 701, 409, 1249, 443

Offset: 1

Robert G. Wilson v, Oct 30 2013

			\r
b\
.2.3...5...7...17...31...73..107..127...257...313...443..1193..1453..1571.=A016041
.3.2..13..23..151..173..233..757..937..1093..1249..1429..1487..1667..1733.=A029971
.4.2...3...5...17...29...59..257..373...409...461...509...787...839...887.=A029972
.5.2...3..31...41...67...83..109..701...911..1091..1171..1277..1327..1667.=A029973
.6.2...3...5....7...37...43...61...67...191...197..1297..1627..1663..1699.=A029974
.7.2...3...5...71..107..157..257..271...307..2549..2647..2801..3347..3697.=A029975
.8.2...3...5....7...73...89...97..113...211...227...251...349...373...463.=A029976
.9.2...3...5....7..109..127..173..191...227...337...373...419...601...619.=A029977
10.2...3...5....7...11..101..131..151...181...191...313...353...373...383.=A002385
11.2...3...5....7..199..277..421..443...499...521...587...643...709...743.=A029978
12.2...3...5....7...11...13..157..181...193...229...241...277...761...773.=A029979
...
inf..2..3..5..7..11..13..17..19..23..29..31..37..41..43..47..53..59..61...=A000040

Cf. A000040, A016041, A029971, A029972, A029973, A029974, A029975, A029976, A029977, A002385, A029978, A029979, A029980, A029981, A029982, A029732, A182231.

A230820 := proc(b,n)
    option remember;
    local a,dgs ;
    if n = 1 then
        if b = 2 then
            return 3;
        else
            return 2;
        end if;
    else
        for a from procname(b,n-1)+1 do
            if isprime(a) then
                ispal := true ;
                dgs := convert(a,base,b) ;
                for i from 1 to nops(dgs)/2 do
                    if op(i,dgs) <> op(-i,dgs) then
                        ispal := false;
                    end if;
                end do:
                if ispal then
                    return a;
                end if;
            end if;
        end do:
    end if;
end proc:
for b from 2 to 9 do
    for n from 1 to 9 do
        printf("%3d ",A230820(b,n)) ;
    end do:
    printf("\n") ;
end do; # R. J. Mathar, Feb 16 2014

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[ n, base]}, idn == Reverse@ idn]; Table[Select[Prime@Range@500, palQ[#, k + 1] &][[b - k + 1]], {b, 11}, {k, b, 1, -1}] // Flatten

cp's OEIS Frontend

A265326 n-th prime minus its binary reversal.

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A046485 Sum of first n palindromic primes A002385.

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A117773 Number of palindromic primes in base 2 with exactly n binary digits.

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A164126 First differences of A006995.

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Python

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A262630 Base-10 representation of 1 and the primes at A262629.

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Mathematica

A333421 Primes that are palindromic in factorial base.

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Mathematica

A056130 Palindromic primes in bases 2 and 4.

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Mathematica

A164125 First differences of A029971.

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