A029973 -id:A029973 - cp's OEIS frontend

A117779 Total number of palindromic primes in base 5 below 5^n.

2, 2, 7, 7, 25, 25, 83, 83, 243, 243, 1038, 1038, 4205, 4205, 17724, 17724, 78553, 78553, 349246, 349246, 1526658, 1526658, 7028772, 7028772, 32055014, 32055014, 146939924, 146939924

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 5) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

Eric Weisstein's World of Mathematics, Palindromic Prime.

Partial sums of A117780.
Cf. A029973, A117700.
Cf. A117772, A117775, A117777, A117781, A117783, A117785, A117787.

a(11)-a(28) from the data at A117780 added by Amiram Eldar, Mar 21 2021

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]]] &]

A117700 Palindromic primes in base 5 (written in base 5).

Original entry on oeis.org

2, 3, 111, 131, 232, 313, 414, 10301, 12121, 13331, 14141, 20102, 20302, 23132, 24342, 31313, 32323, 33133, 34143, 34343, 40304, 41114, 42124, 43334, 44144, 1003001, 1013101, 1111111, 1123211, 1141411, 1221221, 1223221, 1241421, 1303031, 1313131, 1331331, 1411141

Offset: 1

Martin Renner, Apr 13 2006

Eric Weisstein: Palindromic Prime.

Cf. A029973.

Select[FromDigits[IntegerDigits[#, 5]]&/@Prime[Range[3200]], IntegerDigits[#] == Reverse[IntegerDigits[#]]&] (* James C. McMahon, Sep 25 2024 *)

a(26)-a(37) from James C. McMahon, Sep 25 2024

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]] &]

A117780 Total number of palindromic primes in base 5 with n digits.

Original entry on oeis.org

2, 0, 5, 0, 18, 0, 58, 0, 160, 0, 795, 0, 3167, 0, 13519, 0, 60829, 0, 270693, 0, 1177412, 0, 5502114, 0, 25026242, 0, 114884910, 0

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 5) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

Eric Weisstein: Palindromic Prime.

Cf. A029973, A117700.

a(11)-a(28) from Chai Wah Wu, Dec 27 2015

cp's OEIS Frontend

A117779 Total number of palindromic primes in base 5 below 5^n.

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A333421 Primes that are palindromic in factorial base.

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A117700 Palindromic primes in base 5 (written in base 5).

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

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Maple

Mathematica

A333424 Primes that are palindromes in primorial base.

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Mathematica

A117780 Total number of palindromic primes in base 5 with n digits.

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