A002385 -id:A002385 - cp's OEIS frontend

A383816 Palindromic primes which satisfy the requirements of A380943 in at least two ways.

373, 1793971, 7933397, 374636473, 714707417, 727939727, 787333787, 790585097, 947939749, 991999199, 10253935201, 11365556311, 11932823911, 13127372131, 34390609343, 35369996353, 35381318353, 36297179263, 37018281073, 37423332473, 37773537773, 38233333283, 38914541983, 39064546093

Offset: 1

James C. McMahon and Robert G. Wilson v, Jun 09 2025

Terms of A380943 are primes whose decimal representation is the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

			The palindromic prime 373 meets the requirements of A380943 in two ways: the concatenation of 3 and 37 forms the prime 337, and the concatenation of 73 and 3 forms the prime 733.
Although 37673 is a palindrome where 3, 7673, and 76733 are all primes and 3767, 3, and 33767 are all primes, the palindrome is not prime and is therefore not in the sequence.

Subsequence of A383810.

Cf. A000040, A002385, A105184, A380943.

f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[Join[q, p]]; If[ PrimeQ@ FromDigits@ p && PrimeQ@ FromDigits@ q && PrimeQ@ qp && IntegerLength@ qp == len, cnt++]; k++]; cnt]; fQ[n_] := Reverse[idn = IntegerDigits@ n] == idn && f@ n > 1; Select[ Prime@ Range@ 3000000, fQ]

A046349 Composite numbers with only palindromic prime factors.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 144, 147, 150

Offset: 1

Patrick De Geest, Jun 15 1998

Robert Israel, Table of n, a(n) for n = 1..3020

Cf. A002385, A033620, A046350, A046351.

isA046349 := proc(n)
    simplify(isA033620(n) and not isprime(n)) ;
end proc:
for n from 2 to 300 do
    if isA046349(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; Select[Range[4,150],!PrimeQ[#]&&And@@palQ/@First/@FactorInteger[#]&] (* Jayanta Basu, Jun 05 2013 *)
Select[Range[200],CompositeQ[#]&&AllTrue[FactorInteger[#][[All,1]],PalindromeQ]&] (* Harvey P. Dale, May 15 2022 *)

from sympy import isprime, primefactors
def pal(n): s = str(n); return s == s[::-1]
def ok(n): return not isprime(n) and all(pal(f) for f in primefactors(n))
print(list(filter(ok, range(4, 151)))) # Michael S. Branicky, Apr 06 2021

A033620 INTERSECT A002808. - R. J. Mathar, Sep 09 2015

A050251 Number of palindromic primes less than 10^n.

Original entry on oeis.org

0, 4, 5, 20, 20, 113, 113, 781, 781, 5953, 5953, 47995, 47995, 401696, 401696, 3438339, 3438339, 30483565, 30483565, 269577430, 269577430, 2427668363, 2427668363, 22170468927, 22170468927, 202985860292, 202985860292

Offset: 0

Eric W. Weisstein

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, 11. - Martin Renner, Apr 15 2006

Patrick De Geest, World!Of Palindromic Primes, Page 1
Shyam Sunder Gupta, Palindromic Primes up to 10^19.
Shyam Sunder Gupta, Palindromic Primes up to 10^23.
Shyam Sunder Gupta, Palindromic Primes up to 10^25.
Eric Weisstein's World of Mathematics, Palindromic Prime.
Index entries for sequences related to numbers of primes in various ranges

Partial sums of A016115.

Cf. A002113 (palindromes), A002385 (palindromic primes).

from _future_ import division
from sympy import isprime
def paloddgen(l,b=10): # generator of odd-length palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n,n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = divmod(k,b)
                    m = b*m + r
                yield y*n + b*m + k
def A050251(n):
    if n <= 1:
        return 4*n
    else:
        c = 1
        for i in paloddgen((n+1)//2):
            if isprime(i):
                c += 1
        return c # Chai Wah Wu, Jan 05 2015

a(n) ~ A070199(n)/log(10^n) = 1/log(10^n)*Sum {k=1..n} 9*10^floor[(k-1)/2]. - Robert G. Wilson v, May 31 2009

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

More terms from Patrick De Geest, Aug 01 1999
2 more terms from Shyam Sunder Gupta, Feb 12 2006
2 more terms from Shyam Sunder Gupta, Mar 13 2009
a(23)-a(24) from Shyam Sunder Gupta, Oct 05 2013
Missing a(0) inserted by Chai Wah Wu, Nov 21 2021
a(25)-a(26) from Shyam Sunder Gupta, Dec 19 2024

A070250 Palindromic primes with digit sum 10.

Original entry on oeis.org

181, 12421, 30403, 1008001, 1114111, 1212121, 100161001, 100404001, 101060101, 101141101, 102040201, 102202201, 104000401, 130020031, 140000041, 10001610001, 10013031001, 10100600101, 10102220101, 10130003101

Offset: 1

Amarnath Murthy, May 05 2002

Chai Wah Wu, Table of n, a(n) for n = 1..1777

Cf. A002385, A070247, A070248, A070249.

Do[p = IntegerDigits[ Prime[n]]; If[ Plus @@ p == 10 && Reverse[p] == p, Print[ Prime[n]]], {n, 1, 10^10}]
Select[Prime[Range[4607*10^5]],PalindromeQ[#]&&Total[IntegerDigits[#]]==10&] (* Harvey P. Dale, May 28 2023 *)

Edited and extended by Robert G. Wilson v and Jason Earls, May 06 2002

A071251 Squarefree palindromes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 22, 33, 55, 66, 77, 101, 111, 131, 141, 151, 161, 181, 191, 202, 222, 262, 282, 303, 313, 323, 353, 373, 383, 393, 434, 454, 474, 494, 505, 515, 535, 545, 555, 565, 595, 606, 626, 646, 707, 717, 727, 737, 757, 767, 777, 787, 797, 818, 838

Offset: 1

Amarnath Murthy, May 21 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A005117.

Cf. A002385, A069217, A032350, A035132.

ffpal := proc(n) local i,j,k,s,aa,nn,bb,flag; s := n; aa := convert(s,string); nn := length(aa); bb := ``; for i from nn by -1 to 1 do bb := cat(bb,substring(aa,i..i)); od; flag := 0; for j from 1 to nn do if substring(aa,j..j)<>substring(bb,j..j) then flag := 1 fi; od; RETURN(flag); end: ts_ndk_pal := proc(i) if ffpal((i)) = 0 then if (numtheory[issqrfree](i) = 'true' ) then RETURN((i)) fi fi end: andkpal := [seq(ts_ndk_pal(i), i=1..10000)]: andkpal;

Select[ Range[ 1000], # == FromDigits[ Reverse[ IntegerDigits[ # ]]] && Max[ Transpose[ FactorInteger[ # ]] [[2]]] < 2 &]
Select[Range[1000],PalindromeQ[#]&&SquareFreeQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 17 2018 *)

Edited by Robert G. Wilson v, Jun 03 2002

A082623 a(1) = 5, a(n) = smallest palindromic prime obtained by inserting two digits anywhere in a(n-1).

Original entry on oeis.org

5, 151, 10501, 1035301, 103515301, 10325152301, 1013251523101, 101325181523101, 10132512821523101, 1013251428241523101, 101322514282415223101, 10132245142824154223101, 1013224514281824154223101, 101322451402818204154223101, 10132245014028182041054223101

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003

a(78) is the last term, as none of the candidates for a(79) is prime. - Giovanni Resta, Sep 20 2019

Giovanni Resta, Table of n, a(n) for n = 1..78

Cf. A002385, A082620, A082621, A082622, A082624.

cp:= proc(x,y) if x[1] < y[1] then true
           elif x[1] > y[1] then false
           elif nops(x)=1 then true
           else procname(x[2..-1],y[2..-1])
           fi
end proc: A[1]:= 5: L:= [5]:
for n from 2 to 15 do
  nL:= nops(L);
  Lp:= sort([seq(seq([op(L[1..i]), x, op(L[i+1..-1])], x=`if`(i=0, 1..9, 0..9)), i=0..nL)], cp);
  cands:= map(t -> add(t[i]*(10^(i-1)+10^(2*nL+1-i)), i=1..nL)+t[nL+1]*10^(nL), Lp);
  found:= false;
  for i from 1 to nops(cands) do
    if isprime(cands[i]) then
      A[n]:= cands[i];
      L:= Lp[i];
      found:= true;
      break
    fi
  od;
  if not found then break fi
od:
seq(A[i],i=1..15); # Robert Israel, Jan 03 2017, corrected Sep 20 2019

Terms after a(4) corrected by Giovanni Resta, Sep 20 2019

A082624 a(1) = 7, a(n) = smallest palindromic prime obtained by inserting digit anywhere in a(n-1).

Original entry on oeis.org

7, 373, 30703, 3007003, 302070203, 30207570203, 3062075702603, 306020757020603, 30602075057020603, 3060320750570230603, 300603207505702306003, 30060320275057202306003, 3006032021750571202306003, 300603202127505721202306003, 30046032021275057212023064003

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003

Giovanni Resta, Table of n, a(n) for n = 1..315

Cf. A002385, A082620, A082621, A082622, A082623.

More terms from Giovanni Resta, Sep 20 2019

A088284 Palindromic primes in which a single digit is sandwiched between nonempty strings of 9's.

Original entry on oeis.org

919, 929, 99999199999, 99999999299999999, 9999999992999999999, 999999999999919999999999999, 99999999999999499999999999999, 999999999999999999999949999999999999999999999, 99999999999999999999999999899999999999999999999999999

Offset: 1

Amarnath Murthy, Sep 29 2003

Alois P. Heinz, Table of n, a(n) for n = 1..20
Hans Havermann, Near-rep-nine palindromic primes

Subsequence of A002385.

Cf. A088281, A088282, A088283.

Select[Flatten[Table[FromDigits[Join[PadRight[{},n,9],{d},PadRight[{},n,9]]],{n,30},{d,Range[8]}]],PrimeQ] (* Harvey P. Dale, Mar 17 2023 *)

More terms from David Wasserman, Aug 03 2005

Name clarified by Christian Stump, Mar 31 2015

A100580 Palindromic primes containing digits 0 and 1 only. (Palindromic terms in A020449.)

Original entry on oeis.org

11, 101, 100111001, 110111011, 111010111, 1100011100011, 1100101010011, 1101010101011, 1110110110111, 1110111110111, 100110101011001, 101000010000101, 101011000110101, 101110000011101, 110011101110011

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Nov 29 2004

			a(3) = 100111001 because 100111001 is the third palindromic prime composed only of 1's and 0's.

Jon E. Schoenfield, Table of n, a(n) for n = 1..15185 (all terms < 10^38; terms 1..200 from T. D. Noe, 201..2385 from Chai Wah Wu)

Cf. A020449, A002385, A000040, A002113.

PalQ[n_]:=FromDigits[Reverse@IntegerDigits[n]]==n; DeleteDuplicates[Flatten[Table[Select[FromDigits /@ Tuples[{0,1},n],PrimeQ[#]&&PalQ[#] &],{n,15}]]] (* Jayanta Basu, May 11 2013 *)
Select[FromDigits/@Tuples[{0,1},15],PalindromeQ[#]&&PrimeQ[#]&] (* Harvey P. Dale, Jan 18 2023 *)

from sympy import isprime
A100580_list = [11]
for i in range(2, 2**16):
    s = format(i, 'b')
    x = int(s+s[-2::-1])
    if isprime(x):
        A100580_list.append(x) # Chai Wah Wu, Jan 06 2015

A108388 Transmutable primes: Primes with distinct digits d_i, i=1,m (2<=m<=4) such that simultaneously exchanging all occurrences of any one pair (d_i,d_j), i<>j results in a prime.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 179, 191, 199, 313, 331, 337, 773, 911, 919, 1171, 1933, 3391, 7717, 9311, 11113, 11119, 11177, 11717, 11933, 33199, 33331, 77171, 77711, 77713, 79999, 97777, 99991, 113111, 131111, 131113, 131171, 131311

Offset: 1

Rick L. Shepherd, Jun 02 2005

a(n) is a term iff a(n) is prime and binomial(m,2) 'transmutations' (see example) of a(n) are different primes. A083983 is the subsequence for m=2: one transmutation (The author of A083983, Amarnath Murthy, calls the result of such a digit-exchange a self-complement. {Because I didn't know until afterwards that this sequence was a generalization of A083983 and as this generalization always leaves some digits unchanged for m>2, I've chosen different terminology.}). A108389 ({1,3,7,9}) is the subsequence for m=4: six transmutations. Each a(n) corresponding to m=3 (depending upon its set of distinct digits) and having three transmutations is also a member of A108382 ({1,3,7}), A108383 ({1,3,9}), A108384 ({1,7,9}), or A108385 ({3,7,9}). The condition m>=2 only eliminates the repunit (A004022) and single-digit primes. The condition m<=4 is not a restriction because if there were more distinct digits, they would include even digits or the digit 5, in either case transmuting into a composite number. Some terms such as 1933 are reversible primes ("Emirps": A006567) and the reverse is also transmutable. The transmutable prime 3391933 has three distinct digits and is also a palindromic prime (A002385). The smallest transmutable prime having four distinct digits is A108389(0) = 133999337137 (12 digits).

			179 is a term because it is prime and its three transmutations are all prime:
exchanging ('transmuting') 1 and 7: 179 ==> 719 (prime),
exchanging 1 and 9: 179 ==> 971 (prime) and
exchanging 7 and 9: 179 ==> 197 (prime).
(As 791 and 917 are not prime, 179 is not a term of A068652 or A003459 also.).
Similarly, 1317713 is transmutable:
exchanging all 1's and 3s: 1317713 ==> 3137731 (prime),
exchanging all 1's and 7s: 1317713 ==> 7371173 (prime) and
exchanging all 3s and 7s: 1317713 ==> 1713317 (prime).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A108382, A108383, A108384, A108385, A108386, A108389 (transmutable primes with four distinct digits), A083983 (transmutable primes with two distinct digits), A108387 (doubly-transmutable primes), A006567 (reversible primes), A002385 (palindromic primes), A068652 (every cyclic permutation is prime), A003459 (absolute primes).

from gmpy2 import is_prime
from itertools import combinations, count, islice, product
def agen(): # generator of terms
    for d in count(2):
        for p in product("1379", repeat=d):
            p, s = "".join(p), sorted(set(p))
            if len(s) == 1: continue
            if is_prime(t:=int(p)):
                if all(is_prime(int(p.translate({ord(c):ord(d), ord(d):ord(c)}))) for c, d in combinations(s, 2)):
                    yield t
print(list(islice(agen(), 50))) # Michael S. Branicky, Dec 15 2023

cp's OEIS Frontend

A383816 Palindromic primes which satisfy the requirements of A380943 in at least two ways.

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A046349 Composite numbers with only palindromic prime factors.

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A050251 Number of palindromic primes less than 10^n.

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A070250 Palindromic primes with digit sum 10.

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A071251 Squarefree palindromes.

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A082623 a(1) = 5, a(n) = smallest palindromic prime obtained by inserting two digits anywhere in a(n-1).

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A082624 a(1) = 7, a(n) = smallest palindromic prime obtained by inserting digit anywhere in a(n-1).

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A088284 Palindromic primes in which a single digit is sandwiched between nonempty strings of 9's.

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Mathematica

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A100580 Palindromic primes containing digits 0 and 1 only. (Palindromic terms in A020449.)

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