A379354 -id:A379354 - cp's OEIS frontend

A379355 Beginning with 3, least prime such that the reversal concatenation of first n terms is prime.

3, 2, 2, 13, 2, 13, 59, 31, 263, 73, 23, 31, 449, 31, 59, 313, 2, 3, 211, 317, 31, 449, 241, 887, 349, 911, 853, 887, 313, 173, 1777, 179, 967, 503, 331, 113, 163, 359, 1153, 281, 97, 1823, 13, 23, 1657, 269, 223, 3623, 2017, 233, 61, 1361, 367, 1031, 79, 389, 577, 2963, 1741, 59, 13, 1439, 463, 797

Offset: 1

J.W.L. (Jan) Eerland, Dec 21 2024

"Reverse concatenation" here seems to refer to the decimal concatenation R(a(n)) || R(a(n-1)) || ... || R(a(3)) || R(a(2)) || R(a(1)) where R(k) means "reverse digits of k". - N. J. A. Sloane, Jan 03 2025

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A111382, A111383, A113584, A379354.

The primes produced are in A379782.

w = {3};
Do[k = 1;
  q = Monitor[
    Parallelize[
     While[True,
      If[PrimeQ[
         FromDigits[
          Join @@ IntegerDigits /@
            Reverse[
             IntegerDigits[
              FromDigits[
               Join @@ IntegerDigits /@ Append[w, Prime[k]]]]]]], Break[]]; k++];
     Prime[k]], k];
  w = Append[w, q], {i, 2, 57}];
w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    r, an = "", 3
    while True:
        yield int(an)
        r = digits(an)[::-1] + r
        p = 2
        while not is_prime(mpz(digits(p)[::-1]+r)): p = next_prime(p)
        an = p
print(list(islice(agen(), 57))) # Michael S. Branicky, Dec 21 2024

A379761 Beginning with 7, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

7, 3, 3, 31, 389, 1021, 2243, 1831, 5849, 15361, 9887, 3877, 4157, 919, 22637, 14449, 27617, 80221, 5039, 51043, 14009, 126079, 24443, 68311, 49193, 47059, 13049, 253681, 271409, 221227, 138869, 116953, 146297, 21841, 1211549, 322501, 212633, 281791, 216071, 1901749, 38747, 116437

Offset: 1

J.W.L. (Jan) Eerland, Jan 02 2025

			31 is a term because the concatenation of {7,3,3,31} and {13,3,3,7} are respectively 73331 and 13337 which are both prime.
2243 is a term because the concatenation of {7,3,3,31,389,1021,2243} and {3422,1201,983,13,3,3,7} are respectively 7333138910212243 and 3422120198313337 which are both prime.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..100

Cf. A111382, A111383, A113584, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 7: x:= 7:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 7
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 02 2025

A380010 Beginning with 7, least prime such that concatenation of the first n terms is prime.

Original entry on oeis.org

7, 3, 3, 3, 31, 23, 13, 3, 167, 13, 137, 3, 73, 383, 499, 431, 13, 101, 61, 47, 67, 101, 13, 83, 1237, 107, 97, 467, 499, 677, 1423, 353, 73, 431, 331, 683, 487, 2141, 3, 1753, 1787, 31, 443, 139, 653, 1327, 17, 919, 173, 2851, 137, 547, 557, 5167, 347, 7867, 839, 19, 179, 19

Offset: 1

J.W.L. (Jan) Eerland, Jan 09 2025

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..1000

Cf. A111382, A111383, A113584, A379354, A379355, A379761, A380011.

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],k];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, an = "", 7
    while True:
        yield int(an)
        s += digits(an)
        p = 3
        while not is_prime(mpz(s+digits(p))): p = next_prime(p)
        an = p
print(list(islice(agen(), 50))) # after Michael S. Branicky in A379354

A380011 Beginning with 7, least prime such that the reversal concatenation of the first n terms is prime.

Original entry on oeis.org

7, 3, 3, 13, 3, 2, 13, 47, 43, 47, 37, 41, 109, 41, 139, 149, 109, 263, 73, 563, 163, 41, 19, 797, 61, 107, 31, 821, 43, 149, 37, 953, 211, 89, 547, 353, 337, 167, 67, 239, 1009, 449, 97, 23, 349, 41, 31, 911, 61, 929, 229, 797, 331, 191, 463, 107, 463, 809, 2887, 971

Offset: 1

J.W.L. (Jan) Eerland, Jan 09 2025

"Reverse concatenation" here refers to the decimal concatenation R(a(n)) || R(a(n-1)) || ... || R(a(3)) || R(a(2)) || R(a(1)) where R(k) means "reverse digits of k".

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..982

Cf. A111382, A111383, A113584, A379354, A379355, A379761, A380010.

w={7};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]],Break[]];k++];Prime[k]],k];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    r, an = "", 7
    while True:
        yield int(an)
        r = digits(an)[::-1] + r
        p = 2
        while not is_prime(mpz(digits(p)[::-1]+r)): p = next_prime(p)
        an = p
print(list(islice(agen(), 50))) # after Michael S. Branicky in A379355

A202997 a(1) = 3 ; a(n+1) is the prime number obtained by the concatenation {p, a(n)} where p is the smallest prime prefix.

Original entry on oeis.org

3, 23, 223, 31223, 231223, 31231223, 2931231223, 372931231223, 17372931231223, 1317372931231223, 1971317372931231223, 1571971317372931231223, 891571971317372931231223, 79891571971317372931231223, 25179891571971317372931231223, 4325179891571971317372931231223

Offset: 1

Michel Lagneau, Dec 27 2011

By Xylouris' version of Linnik's theorem, a(n) << 3^(6.2^n + cn) for some constant c. [Charles R Greathouse IV, Dec 27 2011]

			a(1) = 3;
a(2) = 23 because 2 is the smallest prime prefix and 23 is prime;
a(3) = 223 because 2 is the smallest prime prefix and 223 is prime;
a(4) = 31223 because 31 is the smallest prime prefix and 31223 is prime.

Cf. A173291, A379354, A379355, A379782.

a0:=3: printf(`%d, `,a0):for it from 1 to 20 do: i:=0:for n from 1 to 1000 while(i=0)  do:p0:=ithprime(n):n0:=length(a0):x:=p0*10^n0+a0: if type(x,prime)=true then printf(`%d, `,x):i:=1:a0:=x:else fi:od:od:

A380227 Beginning with 11, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

11, 3, 11, 31, 59, 463, 131, 103, 599, 3253, 7649, 439, 12791, 2953, 17321, 16651, 10007, 51787, 4871, 1483, 6857, 15649, 53051, 61441, 84449, 35533, 19913, 39097, 23081, 206527, 44939, 189517, 32369, 106657, 606899, 117703, 222977, 220903, 69779, 12007, 95063, 136471, 43973

Offset: 1

J.W.L. (Jan) Eerland, Jan 17 2025

Cf. A113584 (same for 3), A379761 (same for 7).

Cf. A111382, A111383, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 11: x:= 11:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={11};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 11
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # after Michael S. Branicky in A113584

A382898 Beginning with 13, least prime such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

13, 151, 227, 2083, 887, 79, 2963, 1579, 6287, 1321, 6719, 54919, 26699, 8647, 4229, 3919, 102161, 42433, 1667, 192193, 11633, 186343, 47339, 3259, 65963, 14293, 29717, 61297, 28493, 231367, 43793, 145021, 566441, 475903, 92381, 80473, 139967, 882061, 72893, 709279, 6053, 114487, 1179389, 204331, 203351, 139831, 396239, 205327, 501173, 951589

Offset: 1

J.W.L. (Jan) Eerland, Apr 08 2025

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..150

Cf. A113584 (same for 3), A379761 (same for 7), A380227 (same for 11).

Cf. A111382, A111383, A379354, A379355.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(a,b)
  a*10^(1+ilog10(b))+b
end proc:
A:= 13: x:= 13:
for i from 1 to 50 do
   p:= 2:
   do
     p:= nextprime(p);
     y:= tcat(x,p);
     if isprime(y) and isprime(rev(y)) then
          A:= A,p;
          x:= y;
          break
     fi;
   od
od:
A; # after Robert Israel in A113584

w={13};Do[k=1;q=Monitor[Parallelize[While[True,If[PrimeQ[FromDigits[Join@@IntegerDigits/@Reverse[IntegerDigits[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]]]]]&&PrimeQ[FromDigits[Join@@IntegerDigits/@Append[w,Prime[k]]]],Break[]];k++];Prime[k]],{i,k}];w=Append[w,q],{i,2,50}];w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, r, an = "", "", 13
    while True:
        yield int(an)
        d = digits(an)
        s, r, p, sp = s+d, d[::-1]+r, 3, "3"
        while not is_prime(mpz(s+sp)) or not is_prime(mpz(sp[::-1]+r)):
            p = next_prime(p)
            sp = digits(p)
        an = p
print(list(islice(agen(), 40))) # after Michael S. Branicky in A113584

cp's OEIS Frontend

A379355 Beginning with 3, least prime such that the reversal concatenation of first n terms is prime.

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A379761 Beginning with 7, least prime such that concatenation of first n terms and its digit reversal both are primes.

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A380010 Beginning with 7, least prime such that concatenation of the first n terms is prime.

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A380011 Beginning with 7, least prime such that the reversal concatenation of the first n terms is prime.

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A202997 a(1) = 3 ; a(n+1) is the prime number obtained by the concatenation {p, a(n)} where p is the smallest prime prefix.

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A380227 Beginning with 11, least prime such that concatenation of first n terms and its digit reversal both are primes.

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A382898 Beginning with 13, least prime such that concatenation of first n terms and its digit reversal both are primes.

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