A029976 -id:A029976 - cp's OEIS frontend

A117785 Total number of palindromic primes in base 8 below 8^n.

4, 4, 17, 17, 64, 64, 375, 375, 2319, 2319, 15130, 15130, 99554, 99554, 675166, 675166, 4753617, 4753617, 33752394, 33752394, 239605153, 239605153

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 8) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

Eric Weisstein: Palindromic Prime.

Cf. A029976, A006341.

Partial sums of A117786.

revdigs:= proc(n) local L,i;
  L:= convert(n,base,8);
  add(L[-i]*8^(i-1),i=1..nops(L))
end proc:
f:= proc(d) local x,y;
     nops(select(isprime, [seq(seq(x*8^(d+1)+y*8^d+revdigs(x), y=0..7),x=8^(d-1)..8^d-1)]));
end proc:
T:= ListTools:-PartialSums([4, op(map(f,[$1..6]))]):
map(t -> (t,t), T); # Robert Israel, Aug 01 2019

a(9)-a(22) from Robert Israel, Aug 01 2019, using data from A117786.

A006341 Octal palindromes which are also primes.

Original entry on oeis.org

2, 3, 5, 7, 111, 131, 141, 161, 323, 343, 373, 535, 565, 717, 737, 747, 767, 10301, 10601, 11511, 12421, 12621, 13031, 13331, 15151, 16561, 17471, 30403, 30503, 30703, 31313, 31713, 32023, 32323, 33433, 34343, 34443, 35053, 35353, 36463, 36563, 36763, 37473

Offset: 1

James Carlson (carlson(AT)xylogics.com)

Eric Weisstein's World of Mathematics, Palindromic Prime

Cf. A029976.

main(n,i,a,m){while(i=++n) for(a=0; a

More terms from David W. Wilson.

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]]] &]

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]] &]

A117786 Total number of palindromic primes in base 8 with n digits.

Original entry on oeis.org

4, 0, 13, 0, 47, 0, 311, 0, 1944, 0, 12811, 0, 84424, 0, 575612, 0, 4078451, 0, 28998777, 0, 205852759, 0

Offset: 1

Martin Renner, Apr 15 2006

Every palindrome with an even number of digits is divisible by 11 (in base 8) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.

Eric Weisstein: Palindromic Prime.

Cf. A029976, A006341.

a(9)-a(22) from Chai Wah Wu, Dec 25 2015

A168110 Palindromic primes in base 8 which are also emirps (A006567) in base 10.

Original entry on oeis.org

73, 97, 113, 12547, 12611, 13259, 13523, 14107, 14563, 14891, 15667, 15731, 30367, 31799, 31991, 312073, 318281, 350033, 359377, 366169, 371353, 372377, 383833, 392153, 393761, 397921, 792131, 796291, 936227, 936739, 948707, 966379, 992947, 1005427, 1008563, 1029883, 1043899, 1048571, 1311749, 1313797, 1340357, 1358029

Offset: 1

Jonathan Vos Post, Nov 18 2009

What is a good way in the OEIS to show other such pairs of bases analogous to this?

			a(1) = 73 because 73 (base 8) = 111 (which is a palindrome), and R(73) = 37 which is a different prime (base 10). a(2) = 97 because 97 (base 8) = 141 (which is a palindrome), and R(97) = 79 which is a different prime (base 10). a(3) = 113 because 113 (base 8) = 161 (which is a palindrome), and R(113) = 311 which is a different prime (base 10). a(4) = 12547 because 12547 (base 8) = 30403 (which is a palindrome), and R(12547) = 74521 which is a different prime (base 10).

Cf. A000040, A004086, A006567, A007094, A029976.

isA006567 := proc(p) local r; if isprime(p) then r := digrev(p) ; r <> p and isprime(r) ; else false; end if; end proc: isA029803 := proc(n) local dgs,d; dgs := convert(n,base,8) ; for d from 1 to nops(dgs)/2 do if op(d,dgs) <> op(-d,dgs) then return false; end if; end do ; return true; end proc: isA029976 := proc(n) isprime(n) and isA029803(n) ; end proc: isA168110 := proc(p) isA029976(p) and isA006567(p) ; end proc: A168110 := proc(n) option remember ; local a; if n = 1 then 73 ; else a := nextprime(procname(n-1)) ; while not isA168110(a) do a := nextprime(a) ; end do ; return a; end if; end proc: seq(A168110(n),n=1..30) ; # R. J. Mathar, Dec 06 2009

okQ[n_]:=Module[{fridn=FromDigits[Reverse[IntegerDigits[n]]], idn8= IntegerDigits[n,8]}, fridn!=n&&PrimeQ[fridn]&&idn8==Reverse[idn8]]; Select[Prime[Range[75000]],okQ] (* Harvey P. Dale, Aug 10 2011 *)

A029976 INTERSECTION A006567.

Terms beyond a(10) by R. J. Mathar, Dec 06 2009

A046477 Primes that are palindromic in bases 8 and 10.

Original entry on oeis.org

2, 3, 5, 7, 373, 13331, 30103, 1496941, 1970791

Offset: 1

Patrick De Geest, Aug 15 1998

Any other terms have more than 20 digits. - Michael S. Branicky, Dec 19 2020

			373_10 = 565_8. - _Jon E. Schoenfield_, Apr 10 2021

Patrick De Geest, World!Of Palindromic Primes

Cf. A002113, A002385, A029976, A029804.

Do[s = RealDigits[n, 8][[1]]; t = RealDigits[n, 10][[1]]; If[PrimeQ[n], If[FromDigits[t] == FromDigits[Reverse[t]], If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]]], {n, 1, 10^5}]
pal810Q[p_]:=PalindromeQ[p]&&IntegerDigits[p,8]==Reverse[IntegerDigits[p,8]]; Select[ Prime[ Range[150000]],pal810Q] (* Harvey P. Dale, May 25 2023 *)

is(n) = my(d=digits(n, 8), dd=digits(n)); d==Vecrev(d) && dd==Vecrev(dd)
forprime(p=1, , if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Dec 20 2020

# efficiently search to large numbers
from sympy import isprime
from itertools import product
def candidate_prime_pals(digits):
  ruled_out = "024568" # can't be even or multiple of 5
  midrange = [[""], "0123456789"]
  for p in product("0123456789", repeat=digits//2):
    left = "".join(p)
    if len(left):
      if left[0] in ruled_out: continue
    for middle in midrange[digits%2]:
      yield left+middle+left[::-1]
for digits in range(1, 15):
  for p in candidate_prime_pals(digits):
    intp = int(p); octp = oct(intp)[2:]
    if octp==octp[::-1]:
      if isprime(intp):
        print(intp, end=", ") # Michael S. Branicky, Dec 19 2020

# alternate sufficient for producing terms through a(9)
from sympy import isprime
def ispal(n): strn = str(n); return strn==strn[::-1]
for n in range(10**7):
  if ispal(n) and ispal(oct(n)[2:]) and isprime(n):
    print(n) # Michael S. Branicky, Dec 20 2020

A056145 Palindromic primes in bases 2 and 8.

Original entry on oeis.org

3, 5, 7, 73, 28807, 31727, 262657, 295433, 1311749, 1385621, 1478189, 1540157, 1543741, 1549501, 1551037, 1865159, 1932247, 2031599, 2067007, 2085247, 2087807, 83914757, 84663941, 85742021, 85808581, 88779413, 89420117, 89466197, 89924053, 90169301, 94971053, 94983341

Offset: 1

Robert G. Wilson v, Jul 29 2000

Cf. A016041 and A029976.

Do[ If[ PrimeQ[ n ], t=RealDigits[ n, 8 ][ [ 1 ] ]; If[ FromDigits[ t ]==FromDigits[ Reverse[ t ] ], s=RealDigits[ n, 2 ][ [ 1 ] ]; If[ FromDigits[ s ]==FromDigits[ Reverse[ s ] ], Print[ n ] ] ] ], {n, 1, 10^8} ]
Select[Prime[Range[155000]],PalindromeQ[IntegerDigits[#,2]] && PalindromeQ[ IntegerDigits[ #,8]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 23 2017 *)

More terms from Harvey P. Dale, Sep 23 2017

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

Robert G. Wilson v, Jul 29 2000

Cf. A029972 and A029976.

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}]

cp's OEIS Frontend

A117785 Total number of palindromic primes in base 8 below 8^n.

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A006341 Octal palindromes which are also primes.

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C

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A333421 Primes that are palindromic in factorial base.

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Mathematica

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

A117786 Total number of palindromic primes in base 8 with n digits.

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A168110 Palindromic primes in base 8 which are also emirps (A006567) in base 10.

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Mathematica

Formula

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A046477 Primes that are palindromic in bases 8 and 10.

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