A113584 -id:A113584 - cp's OEIS frontend

A111382 Beginning with 3, least number such that concatenation of first n terms and its digit reversal both are primes.

3, 1, 1, 21, 11, 43, 47, 157, 753, 51, 917, 273, 2409, 703, 413, 3729, 1153, 6243, 8789, 2307, 4477, 137, 403, 10649, 4617, 4533, 6133, 4721, 877, 2469, 5967, 1557, 1047, 38931, 15533, 6877, 23987, 4767, 18049, 1463, 118333, 27897

Offset: 1

Hans Havermann, Nov 08 2005

Robert Israel, Table of n, a(n) for n = 1..160

Cf. A113584.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc;
R:= 3: X:= 3: XR:= 3:
for i from 2 to 50 do
  for x from 1 by 2 do
    d:= 1+ilog10(x);
    t:= X*10^(1+ilog10(x)) + x;
    if not isprime(t) then next fi;
    xr:= rev(x);
    tr:= XR+xr*10^(1+ilog10(XR));
    if isprime(tr) then break fi;
  od;
  X:= t; XR:= tr; R:= R,x;
od:
R; # Robert Israel, Aug 09 2023

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz
def agen(): # generator of terms
    s, r, an = "", "", 3
    while True:
        yield int(an)
        d = digits(an)
        s, r, k, sk = s+d, d[::-1]+r, 1, "1"
        while not is_prime(mpz(s+sk)) or not is_prime(mpz(sk[::-1]+r)):
            k += 2
            if k%10 == 5: k += 2
            sk = digits(k)
        an = k
print(list(islice(agen(), 42))) # Michael S. Branicky, Jan 02 2025

A111383 Beginning with 3, least member of A007500 such that concatenation of first n terms and its digit reversal both are primes.

Original entry on oeis.org

3, 7, 3, 3, 79, 701, 157, 1103, 11959, 1901, 10273, 92753, 17047, 11909, 144973, 327251, 99289, 92831, 90373, 309671, 1149619, 745397, 1232083, 94793, 18481, 76607, 186649, 181421, 1657561, 3746111, 7067239, 324143, 3185263, 9457181, 1703413, 3517583, 72481, 12859481

Offset: 1

Hans Havermann, Nov 08 2005

Michael S. Branicky, Table of n, a(n) for n = 1..164

Cf. A113584, A111382, A007500.

from gmpy2 import digits, is_prime, mpz
from itertools import count, islice, product
def bgen(): # generator of terms of A007500 -{2, 5} as strings
    yield from "37"
    p = 11
    for digits in count(2):
        for first in "1379":
            for mid in product("0123456789", repeat=digits-2):
                for last in "1379":
                    s = first + "".join(mid) + last
                    if is_prime(mpz(s)) and is_prime(mpz(s[::-1])):
                        yield s
def agen(): # generator of terms
    s, r, an, san = "", "", 3, "3"
    while True:
        yield int(an)
        s, r = s+san, san[::-1]+r
        for san in bgen():
            if is_prime(mpz(s+san)) and is_prime(mpz(san[::-1]+r)):
                break
        an = mpz(san)
print(list(islice(agen(), 34))) # Michael S. Branicky, Jan 02 2025

a(35) and beyond from Michael S. Branicky, Jan 02 2025

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

Original entry on oeis.org

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

A379354 Beginning with 3, least prime such that concatenation of first n terms is prime.

Original entry on oeis.org

3, 7, 3, 3, 7, 29, 43, 11, 61, 71, 19, 191, 43, 53, 7, 239, 31, 173, 43, 137, 79, 53, 13, 557, 619, 47, 271, 797, 463, 83, 211, 467, 229, 131, 199, 359, 1249, 887, 853, 641, 109, 257, 1153, 1031, 613, 953, 607, 641, 499, 359, 1297, 1031, 2137, 401, 283, 29, 1321, 1499, 547, 83, 397, 2153, 1759, 1277

Offset: 1

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

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

Cf. A111382, A111383, A113584, A379355.

w = {3};
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, 57}];
w

from itertools import count, islice
from gmpy2 import digits, is_prime, mpz, next_prime
def agen(): # generator of terms
    s, an = "", 3
    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(), 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

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

A111382 Beginning with 3, least number such that concatenation of first n terms and its digit reversal both are primes.

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A111383 Beginning with 3, least member of A007500 such that concatenation of first n terms and its digit reversal both are primes.

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

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A379354 Beginning with 3, least prime such that 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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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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