A002385 -id:A002385 - cp's OEIS frontend

A039954 Palindromic primes formed from the reflected decimal expansion of Pi.

3, 313, 31415926535897932384626433833462648323979853562951413

Offset: 1

Carlos Rivera reports that the next two members of this sequence have 301 and 921 digits. The first has been tested with APRTE-CLE. The second one is only a StrongPseudoPrime at the moment. - May 16 2003
Thomas Spahni reports that the fifth member of this sequence with 921 digits is prime. He used Francois Morain's ECPP-V6.4.5a which proved primality in 14913.7 seconds running on a Celeron Core2 CPU at 2.00GHz. - Jun 05 2008
Primes in A135697. Terms with an odd number of digits are the primes in A135698. - Omar E. Pol, Mar 06 2012

C. K. Caldwell and G. L. Honaker, Jr., Prime Curios!, 31414...51413 (53-digits)
Shyam Sunder Gupta, Mystery of pi, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Cf. A002385, A119351, A135697, A135698.

Select[Table[p = Flatten[RealDigits[Pi, 10, d]]; (FromDigits[p] - 1)*10^(Length[p] - 3) + FromDigits[Drop[Reverse[p], 2]], {d, 27}], PrimeQ] (* Arkadiusz Wesolowski, Dec 18 2011 *)

A084092 Prime power decimal palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 101, 121, 131, 151, 181, 191, 313, 343, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1331, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14641, 14741, 15451, 15551, 16061, 16361

Offset: 1

Reinhard Zumkeller, May 11 2003

			121=A000961(42)=A002113(21), therefore 121 is a term;
131=A000961(46)=A002113(22), therefore 131 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..600
Eric Weisstein's World of Mathematics, Palindromic Prime.
Eric Weisstein's World of Mathematics, Prime Power.

Intersection of A000961 and A002113, union of A002385 and A084093.

Join[{1}, Select[Range[16370], Reverse[x = IntegerDigits[#]] == x && PrimePowerQ[#] &]] (* Jayanta Basu, Jun 24 2013 *)
Join[{1},Select[Range[17000],PalindromeQ[#]&&PrimePowerQ[#]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 07 2019 *)

Corrected and extended by T. D. Noe, Oct 25 2006

A117697 Palindromic primes in base 2 (written in base 2).

Original entry on oeis.org

11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, 10010101001, 10110101101, 11000100011, 11001010011, 11011111011, 11100100111, 11101010111, 1001100011001, 1001111111001, 1010001000101

Offset: 1

Martin Renner, Apr 13 2006

Attila Olah, Table of n, a(n) for n=1..43964
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Cf. A016041, A002385.

isA016041 := proc(n)
    local bin,dig ;
    if isprime(n) then
        bin := convert(n,base,2) ;
        for dig from 1 to nops(bin)/2 do
            if op(dig,bin) <> op(-dig,bin) then
                return false;
            end if;
        end do ;
        return true;
    else
        false ;
    end if ;
end proc:
for i from 1 to 900 do p := ithprime(i) : if isA016041(p) then printf("%d, ",A007088(p)) ; fi ; od : # R. J. Mathar, Feb 25 2007

pal2Q[n_] := Reverse[x = IntegerDigits[n, 2]] == x; BaseForm[Select[Prime[Range[700]], pal2Q[#] &], 2] (* Jayanta Basu, Jun 24 2013 *)
FromDigits /@ Select[IntegerDigits[Prime@ Range[1000], 2], PalindromeQ] (* Michael De Vlieger, Oct 28 2020 *)

a(n) = A007088(A016041(n)). - R. J. Mathar, Feb 25 2007

A037010 Differences between adjacent palindromic primes.

Original entry on oeis.org

1, 2, 2, 4, 90, 30, 20, 30, 10, 122, 40, 20, 10, 344, 30, 30, 10, 122, 10, 9372, 200, 100, 710, 100, 1010, 300, 100, 510, 500, 100, 410, 400, 710, 100, 510, 300, 200, 100, 810, 500, 210, 300, 910, 500, 100, 10112

Offset: 1

G. L. Honaker, Jr.

			E.g. 11 - 7 = 4.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
P. De Geest, World!Of Palindromic Primes

Cf. A002385, A327427 (records).

Differences[Select[Prime[Range[3260]], Reverse[x = IntegerDigits[#]] == x &]] (* Jayanta Basu, Jun 24 2013 *)

a(n) = A002385(n+1) - A002385(n). - Michel Marcus, Sep 12 2019

A069217 Numbers n such that phi(n) + sigma(n) = n + reversal(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181

Offset: 1

Joseph L. Pe, Apr 11 2002

Note that all terms so far are palindromes.
It is obvious that if n is a term of the sequence greater than 1 then n is prime iff n is a palindrome. Do there exist composite terms in the sequence? - Farideh Firoozbakht, Jan 28 2006 Answer: Yes, see next comment.
Giovanni Resta writes (Sep 06 2006): The smallest composite number such that n+rev(n)=phi(n)+sigma(n) is n = 3197267223 = 3 * 79 * 677 * 19927 with rev(n) = 3227627913, phi(n) = 2101316256, sigma(n) = 4323578880 and 3197267223+3227627913 = 6424895136 = 2101316256+4323578880.

			phi(101) + sigma(101) = 202 = 101 + 101 = 101 + reversal(101).

Vincenzo Librandi, Table of n, a(n) for n = 1..700

Contains composite terms, so is strictly different from A002385.

Select[Range[5*10^4], EulerPhi[ # ] + DivisorSigma[1, # ] == # + FromDigits[Reverse[IntegerDigits[ # ]]] &]

If p is prime and rev(p)=p then p+rev(p)=2p=phi(p)+sigma(p) so all palindromic primes are in the sequence. - Farideh Firoozbakht, Sep 12 2006

A016115 Number of prime palindromes with n digits.

Original entry on oeis.org

4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, 42042, 0, 353701, 0, 3036643, 0, 27045226, 0, 239093865, 0, 2158090933, 0, 19742800564, 0, 180815391365, 0

Offset: 1

Robert G. Wilson v

Every palindrome with an even number of digits is divisible by 11 and therefore is composite (not prime). Hence there is only one palindromic prime with an even number of digits, namely 11 itself. - Martin Renner, Apr 15 2006

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

K. S. Brown, On General Palindromic Numbers
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 36.
Patrick De Geest, World!Of Palindromic Primes
Shyam Sunder Gupta, Palindromic Primes up to 10^19, Feb. 6, 2006.
Shyam Sunder Gupta, Palindromic Primes up to 10^21, March 13, 2009.
Shyam Sunder Gupta, Palindromic Primes up to 10^23, Oct. 4, 2013.
Shyam Sunder Gupta, Palindromic Primes up to 10^25, Dec. 18, 2024.
Eric Weisstein's World of Mathematics, Palindromic Prime.

Cf. A002113 (palindromes), A002385 (palindromic primes), A040025 (bisection), A050251 (partial sums).

# A016115 Gets numbers of base-10 palindromic primes with exactly d digits, 1 <= d <= 13 (say), in the list "lis"
lis:=[4,1];
for d from 3 to 13 do
if d::even then
    lis:=[op(lis),0];
else
    m:= (d-1)/2:
    Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
    ct:=0; for x in Res2 do if isprime(x) then ct:=ct+1; fi: od:
    lis:=[op(lis),ct];
fi:
lprint(d,lis);
od:
lis; # N. J. A. Sloane, Oct 18 2015

A016115[n_] := Module[{i}, If[EvenQ[n] && n > 2, Return[0]]; Return[Length[Select[Range[10^(n - 1), 10^n - 1], # == IntegerReverse[#] && PrimeQ[#] &]]]];
Table[A016115[n], {n, 6}] (* Robert Price, May 25 2019 *)
(* -OR-  A less straightforward implementation, but more efficient in that the palindromes are constructed instead of testing every number in the range. *)
A016115[n_] := Module[{c, f, t0, t1},
   If[n == 2, Return[1]];
   If[EvenQ[n], Return[0]];
   c = 0; t0 = 10^((n - 1)/2); t1 = t0*10;
   For[f = t0, f < t1, f++,
    If[n != 1 && MemberQ[{2,4,5,6,8}, Floor[f/t0]], f = f + t0 - 1; Continue[]];
    If[PrimeQ[f*t0 + IntegerReverse[Floor[f/10]]], c++]]; Return[c]];
Table[A016115[n], {n, 1, 12}] (* Robert Price, May 25 2019 *)

apply( {A016115(n)=if(n%2, (n<3)+vecsum([sum(k=i, i+n, (k*2-k%10)%3 && isprime(k*n+fromdigits(Vecrev(digits(k\10))))) | i<-[1, 3, 7, 9]*n=10^(n\2)]), n==2)}, [1..12]) \\ M. F. Hasler, Dec 19 2024

from sympy import isprime
from itertools import product
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def a(n): return int(n==2) if n%2 == 0 else sum(isprime(p) for p in pals(n))
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Jun 23 2021

a(2n) = 0 for n > 1. - Chai Wah Wu, Nov 21 2021

Corrected and extended by Patrick De Geest, Jun 15 1998
a(17) = 27045226 was found by Martin Eibl (M.EIBL(AT)LINK-R.de) and independently by Warut Roonguthai and later confirmed by Carlos Rivera, in June 1998.
a(19) from Shyam Sunder Gupta, Feb 12 2006
a(21)-a(22) from Shyam Sunder Gupta, Mar 13 2009
a(23)-a(24) from Shyam Sunder Gupta, Oct 05 2013
a(25)-a(26) from Shyam Sunder Gupta, Dec 19 2024

A229875 Iterated sum-of-digits of palindromic prime; or digital root of palindromic prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 2, 5, 7, 1, 2, 7, 2, 4, 5, 7, 1, 4, 5, 1, 2, 5, 7, 8, 7, 8, 1, 4, 5, 2, 7, 8, 4, 8, 7, 8, 5, 8, 1, 2, 2, 7, 1, 4, 5, 1, 2, 7, 8, 1, 4, 5, 8, 4, 4, 5, 8, 1, 4, 7, 8, 1, 5, 2, 5, 4, 7, 4, 5, 2, 8, 7, 1, 2, 1, 7, 2, 7, 2, 4, 8, 4, 2, 2, 2, 5, 4

Offset: 1

Shyam Sunder Gupta, Oct 02 2013

Integers with digital root 3, 6 or 9 are divisible by 3, so 3 is the only palindromic prime with digital root 3 and there are no palindromic primes with digital root 6 or 9.

			a(7)=5 because the 7th palindromic prime is 131 and 1+3+1 = 5.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..5953

Cf. A002385, A010888, A038194.

t = {}; Do[z = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[z], AppendTo[t, Mod[z, 9]]], {n, 1, 99999}]; Insert[t, 2, 5]
Mod[#,9]&/@Select[Prime[Range[9000]],PalindromeQ] (* Harvey P. Dale, Mar 25 2025 *)

a(n) = A010888(A002385(n)). - R. J. Mathar, Sep 09 2015

A256076 Non-palindromic balanced primes.

Original entry on oeis.org

1823, 1933, 2141, 2251, 2633, 2963, 3061, 3391, 4091, 4253, 4363, 4583, 5393, 5717, 5827, 6637, 6857, 6967, 7829, 8147, 8419, 8969, 9067, 9397, 14303, 14503, 15013, 15313, 15413, 15913, 16223, 16823, 17033, 17333, 18043, 18143, 18443, 18743, 19553, 19753, 19853

Offset: 1

Eric Angelini and M. F. Hasler, Mar 14 2015

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero. Palindromic primes (A002385) are "trivially" balanced, so they are excluded here.
These are the primes in A256075, see there for further information.
See A256081 for the binary version and A256090 for the hexadecimal version.

			a(1)=1823 is balanced because 1*3/2 + 8*1/2 = 2*1/2 + 3*3/2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A256075, A256081, A256090, A002385.

filter:= proc(n) local L,m;
  L:= convert(n,base,10);
  m:= (1+nops(L))/2;
add(L[i]*(i-m),i=1..nops(L))=0  and isprime(n) and L <> ListTools:-Reverse(L)
end proc:
select(filter, [seq(i,i=1001..20000,2)]); # Robert Israel, May 29 2018

is(n,d=digits(n),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)&&isprime(n)

A045336 Palindromic terms from A019546.

Original entry on oeis.org

2, 3, 5, 7, 353, 373, 727, 757, 32323, 33533, 35353, 35753, 37273, 37573, 72227, 72727, 73237, 75557, 77377, 3222223, 3223223, 3233323, 3252523, 3272723, 3337333, 3353533, 3553553, 3722273, 3732373, 3773773, 7257527, 7327237, 7352537, 7527257, 7722277

Offset: 1

Robert G. Wilson v, Aug 18 2000

a(33) = 7352537 is the smallest palindromic prime using all prime digits (see Prime Curios! link). - Bernard Schott, Nov 10 2020

Michael S. Branicky, Table of n, a(n) for n = 1..12725 (all terms with <= 17 digits; terms 1..330 from Harvey P. Dale)
Chris K. Caldwell and G. L. Honaker, Jr., 7352537, Prime Curios!

Cf. A019546 and A002385.

Select[ Range[ 1, 10^7 ], PrimeQ[ # ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 0 ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 1 ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 4 ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 6 ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 8 ] && FreeQ[ RealDigits[ # ][ [ 1 ] ], 9 ] && RealDigits[ # ][ [ 1 ] ] == Reverse[ RealDigits[ # ][ [ 1 ] ] ] & ]
Table[FromDigits/@Select[Tuples[{2,3,5,7},n],#==Reverse[#]&&PrimeQ[ FromDigits[ #]]&],{n,12}]//Flatten (* Harvey P. Dale, Jun 19 2016 *)
Select[Flatten[Table[FromDigits/@Tuples[{2,3,5,7},n],{n,10}]],PrimeQ[#]&&PalindromeQ[#]&] (* Harvey P. Dale, Mar 24 2025 *)
f@n_ := Prime@n;
g@l_ := FromDigits@# & /@ Table[Join[l, {f@i}, Reverse@l], {i, 4}];
Flatten[g@# & /@ (f@# & /@
     Select[Table[IntegerDigits[n, 5], {n, 2000}], FreeQ[#, 0] &])] //
Select[PrimeQ] (* Hans Rudolf Widmer, Dec 18 2021 *)

from sympy import isprime
from itertools import count, product, takewhile
def primedigpals():
    for d in count(1, 2):
        for p in product("2357", repeat=d//2):
            left = "".join(p)
            for mid in "2357":
                yield int(left + mid + left[::-1])
def aupto(N):
    return list(takewhile(lambda x: x<=N, filter(isprime, primedigpals())))
print(aupto(10**7)) # Michael S. Branicky, Dec 18 2021

A046368 Products of two palindromic primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 262, 302, 303, 362, 382, 393, 453, 505, 543, 573, 626, 655, 706, 707, 746, 755, 766, 905, 917, 939, 955, 1057, 1059, 1111, 1119, 1149, 1267, 1337, 1441, 1454, 1514, 1565, 1574, 1594, 1661, 1765, 1838, 1858

Offset: 1

Patrick De Geest, Jun 15 1998

Intersection of A001358 and A033620; A046368 is a subsequence. - Reinhard Zumkeller, Apr 10 2011
Equivalently, semiprimes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
See A046376 for the subsequence of palindromic terms. - M. F. Hasler, Jan 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes), A046376 (subsequence of palindromes), A046400.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Range[1800], PrimeOmega[#] == 2 && And @@ palQ /@ First /@ FactorInteger[#] &] (* Jayanta Basu, Jun 23 2013 *)

select( {is_A046368(n)=bigomega(n)==2 && vecmin( apply( is_A002113, factor(n)[, 1]))}, [1..9999]) \\ M. F. Hasler, Jan 04 2022

Definition clarified by Franklin T. Adams-Watters, Apr 11 2011

Definition simplified by M. F. Hasler, Jan 04 2022

cp's OEIS Frontend

A039954 Palindromic primes formed from the reflected decimal expansion of Pi.

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A084092 Prime power decimal palindromes.

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A117697 Palindromic primes in base 2 (written in base 2).

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A037010 Differences between adjacent palindromic primes.

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A069217 Numbers n such that phi(n) + sigma(n) = n + reversal(n).

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A016115 Number of prime palindromes with n digits.

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A229875 Iterated sum-of-digits of palindromic prime; or digital root of palindromic prime.

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A256076 Non-palindromic balanced primes.

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