A029971 -id:A029971 - cp's OEIS frontend

A164125 First differences of A029971.

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

A016041 Primes that are palindromic in base 2 (but written here in base 10).

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191, 17377, 18097, 18289, 19433, 19609, 19801, 21157, 22541, 22669, 22861, 23581, 24029

Offset: 1

Robert G. Wilson v

See A002385 for palindromic primes in base 10, and A256081 for primes whose binary expansion is "balanced" (see there) but not palindromic. - M. F. Hasler, Mar 14 2015

Number of terms less than 4^k, k=1,2,3,...: 1, 3, 5, 8, 11, 18, 30, 53, 93, 187, 329, 600, 1080, 1936, 3657, 6756, 12328, 23127, 43909, 83377, 156049, 295916, 570396, 1090772, 2077090, 3991187, 7717804, 14825247, 28507572, 54938369, 106350934, ..., partial sums of A095741 plus 1. - Robert G. Wilson v, Feb 23 2018, corrected by Jeppe Stig Nielsen, Jun 17 2023

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov, terms 1001..3000 from Michael De Vlieger)
Kevin S. Brown, On General Palindromic Numbers
Patrick De Geest, World!Of Palindromic Primes

Intersection of A000040 and A006995.
First row of A095749.
A095741 gives the number of terms in range [2^(2n), 2^(2n+1)].
Cf. A095730 (primes whose Zeckendorf expansion is palindromic), A029971 (primes whose ternary (base-3) expansion is palindromic).
Cf. A117697 (written in base 2), A002385, A194097, A256081.

[NthPrime(n): n in [1..5000] | (Intseq(NthPrime(n), 2) eq Reverse(Intseq(NthPrime(n), 2)))]; // Vincenzo Librandi, Feb 24 2018

lst = {}; Do[ If[ PrimeQ@n, t = IntegerDigits[n, 2]; If[ FromDigits@t == FromDigits@ Reverse@ t, AppendTo[lst, n]]], {n, 3, 50000, 2}]; lst (* syntax corrected by Robert G. Wilson v, Aug 10 2009 *)
pal2Q[n_] := Reverse[x = IntegerDigits[n, 2]] == x; Select[Prime[Range[2800]], pal2Q[#] &] (* Jayanta Basu, Jun 23 2013 *)
genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 0; lst = {}; While[k < 100, AppendTo[lst, Select[ genPal[k, 2], PrimeQ]]; lst = Flatten@ lst; k++]; lst (* Robert G. Wilson v, Feb 23 2018 *)

is(n)=isprime(n)&&Vecrev(n=binary(n))==n \\ M. F. Hasler, Feb 23 2018

from sympy import isprime
def ok(n): return isprime(n) and (b:=bin(n)[2:]) == b[::-1]
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 20 2024

Sum_{n>=1} 1/a(n) = A194097. - Amiram Eldar, Mar 19 2021

More terms from Patrick De Geest

A117698 Palindromic primes in base 3 (written in base 3).

Original entry on oeis.org

2, 111, 212, 12121, 20102, 22122, 1001001, 1021201, 1111111, 1201021, 1221221, 2001002, 2021202, 2101012, 2111112, 2121212, 2201022, 2211122, 102111201, 110111011

Offset: 1

Martin Renner, Apr 13 2006

R. J. Mathar, Table of n, a(n) for n = 1..179
Eric Weisstein: Palindromic Prime.

Cf. A029971.

A117775 Total number of palindromic primes in base 3 below 3^n.

Original entry on oeis.org

1, 1, 3, 3, 6, 6, 18, 18, 26, 26, 73, 73, 179, 179, 459, 459, 1179, 1179, 3004, 3004, 8111, 8111, 22183, 22183, 60789, 60789, 168641, 168641, 469689, 469689, 1322664, 1322664, 3691761, 3691761, 10390938, 10390938, 29502559, 29502559, 84012658, 84012658, 239417332, 239417332

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 3) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

			a(5) = a(6) = 6 as the six palindromic primes below 3^5 are 2_10 = 2_3, 13_10 = 111_3, 23_10 = 212_3, 151_10 = 12121_3, 173_10 = 20102_3, 233_10 = 22122_3. There are no palindromic primes with 6 digits so a(5) = a(6). - _David A. Corneth_, Mar 21 2021

Eric Weisstein's World of Mathematics, Palindromic Prime.

Partial sums of A117776.
Cf. A029971, A117698.
Cf. A117772, A117777, A117779, A117781, A117783, A117785, A117787.

a(2*k-1) = a(2*k) for k >= 1. - Bernard Schott, Mar 23 2021

a(15)-a(42) from the data at A117776 added by Amiram Eldar, Mar 21 2021

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

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]]] &]

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

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

A333424 Primes that are palindromes in primorial base.

Original entry on oeis.org

3, 7, 11, 31, 47, 211, 223, 229, 281, 293, 2311, 2347, 2383, 2843, 2879, 30091, 30181, 30211, 30307, 30367, 30427, 30493, 30553, 30643, 30829, 30859, 34871, 34961, 35051, 35117, 35267, 35363, 35393, 35423, 510751, 511711, 513067, 513307, 515143, 517459, 518179

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			3 is a term since it is a prime number and its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..300
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Intersection of A000040 and A333423.
Cf. A049345, A235168.
Cf. A002385, A016041, A029971, A029972, A029973, A029974, A029975, A029976, A029977, A029978, A029979, A029980, A029981, A029982, A333421.

max = 8; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[nmax], PrimeQ[#] && PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

A117776 Total number of palindromic primes in base 3 with n digits.

Original entry on oeis.org

1, 0, 2, 0, 3, 0, 12, 0, 8, 0, 47, 0, 106, 0, 280, 0, 720, 0, 1825, 0, 5107, 0, 14072, 0, 38606, 0, 107852, 0, 301048, 0, 852975, 0, 2369097, 0, 6699177, 0, 19111621, 0, 54510099, 0, 155404674, 0

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 3) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

Eric Weisstein: Palindromic Prime.

Cf. A029971, A117698.

a(15)-a(42) from Chai Wah Wu, Dec 27 2015

cp's OEIS Frontend

A164125 First differences of A029971.

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A016041 Primes that are palindromic in base 2 (but written here in base 10).

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PARI

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A117698 Palindromic primes in base 3 (written in base 3).

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A117775 Total number of palindromic primes in base 3 below 3^n.

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A164126 First differences of A006995.

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A333421 Primes that are palindromic in factorial base.

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A164124 First differences of A016041.

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

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Maple