cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A117777 Total number of palindromic primes in base 4 below 4^n.

Original entry on oeis.org

2, 3, 6, 6, 16, 16, 45, 45, 142, 142, 435, 435, 1417, 1417, 4802, 4802, 16433, 16433, 57808, 57808, 210708, 210708, 752233, 752233, 2748264, 2748264, 10211513, 10211513, 38013234, 38013234, 140417288, 140417288
Offset: 1

Views

Author

Martin Renner, Apr 15 2006

Keywords

Comments

Every palindrome with an even number of digits is divisible by 11 (in base 4) and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits.

Links

Crossrefs

Partial sums of A117778.
Cf. A029972, A117699.
Cf. A117772, A117775, A117779, A117781, A117783, A117785, A117787.

Extensions

a(13)-a(32) from the data at A117778 added by Amiram Eldar, Mar 21 2021

A117699 Palindromic primes in base 4 (written in base 4).

Original entry on oeis.org

2, 3, 11, 101, 131, 323, 10001, 11311, 12121, 13031, 13331, 30103, 31013, 31313, 32023, 33133, 1003001, 1013101, 1101011, 1120211, 1123211, 1133311, 1201021, 1212121, 1221221, 1302031, 1311131, 1321231
Offset: 1

Views

Author

Martin Renner, Apr 13 2006

Keywords

Links

Crossrefs

Cf. A029972.

A262634 Base-10 representation of 0 and the primes at A262633.

Original entry on oeis.org

0, 17, 839, 15647, 19875809, 271854991487, 1037745510031339, 1723740375514348277, 79540916124198230488513, 5823206408678014785594704353, 331079896899961216174255077712193, 5255024346804703767393892624827437183, 679574531660517571339207845002833364989
Offset: 1

Views

Author

Clark Kimberling, Oct 02 2015

Keywords

Examples

			n   A262633(n)    base-10 representation
1   0                0
2   101              17
3   31013            839

Links

Crossrefs

Cf. A262633. Subsequence of A029972.

Programs

  • Mathematica
    s = {0}; base = 4; 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  (* A262633 *)
Map[FromDigits[ToString[#], base] &, s]  (* A262634 *)
(* Peter J. C. Moses, Sep 01 2015 *)

A262636 Base-10 representation of 1 and the primes at A262635.

Original entry on oeis.org

1, 409, 88469, 3499607, 525755773, 1889074241401, 14476124563693171, 3690382506839631667, 944489835647492584243, 17945058791198905978063, 1798278215177468964031733, 17136765345945336793560989047, 39126270988987383599688999460333
Offset: 1

Views

Author

Clark Kimberling, Oct 02 2015

Keywords

Examples

			n   A262635(n)    base-10 representation
1   1             1
2   12121         409
3   111212111     88469

Links

Crossrefs

Cf. A262635. Subsequence of A029972.

Programs

  • Mathematica
    s = {1}; base = 4; 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  (* A262635 *)
Map[FromDigits[ToString[#], base] &, s]  (* A262636 *)
(* 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

Views

Author

Amiram Eldar, Mar 20 2020

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Robert G. Wilson v, Jul 29 2000

Keywords

Comments

Intersection of A016041 and A029972.
Subsequence of primes of A097856. - Michel Marcus, Nov 07 2015

Examples

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

Links

Crossrefs

Cf. A016041 and A029972.

Programs

  • Mathematica
    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}]

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

Views

Author

Robert G. Wilson v, Oct 30 2013

Keywords

Examples

			\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

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Amiram Eldar, Mar 20 2020

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

A117778 Total number of palindromic primes in base 4 with n digits.

Original entry on oeis.org

2, 1, 3, 0, 10, 0, 29, 0, 97, 0, 293, 0, 982, 0, 3385, 0, 11631, 0, 41375, 0, 152900, 0, 541525, 0, 1996031, 0, 7463249, 0, 27801721, 0, 102404054, 0
Offset: 1

Views

Author

Martin Renner, Apr 15 2006

Keywords

Comments

Every palindrome with an even number of digits is divisible by 11 (in base 4) and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits.

Links

Crossrefs

Cf. A029972, A117699.

Extensions

a(13)-a(32) from Chai Wah Wu, Dec 27 2015

A056146 Palindromic primes in bases 4 and 8.

Original entry on oeis.org

2, 3, 5, 373, 4289, 6761, 14891, 15083, 1311749, 1313797, 1444901, 1530229, 1531253, 17480321, 17563841, 17730241, 17734337, 27263017, 27533417, 27957929, 28053737, 50339843, 50506243, 50856067, 51122371, 61254251
Offset: 1

Views

Author

Robert G. Wilson v, Jul 29 2000

Keywords

Crossrefs

Cf. A029972 and A029976.

Programs

  • Mathematica
    Do[ If[PrimeQ[n], t = RealDigits[n, 8][[1]]; If[FromDigits[t] == FromDigits[Reverse[t]], s = RealDigits[n, 4][[1]]; If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]]], {n, 1, 10^8, 2}]
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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