A034821 -id:A034821 - cp's OEIS frontend

A034808 Concatenation of 'prevprime(k) and k' is a prime.

3, 9, 37, 39, 51, 63, 87, 89, 111, 117, 123, 153, 157, 163, 173, 177, 183, 207, 211, 213, 217, 219, 239, 249, 257, 263, 267, 269, 273, 277, 279, 289, 321, 323, 327, 333, 337, 339, 343, 359, 369, 379, 407, 423, 439, 441, 459, 471, 473, 477, 479, 489, 497, 513

Offset: 1

Patrick De Geest, Oct 15 1998

Since there are primes in the sequence, and concat(p,p) = p*(10^x+1) is always composite, it is clear that here the variant 2 (A151799(n) < n) of the prevprime function is used, rather than the variant 1 (A007917(n) <= n). - M. F. Hasler, Sep 09 2015

			n=333 -> previous prime is 331, thus '331333' is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034809-A034821.

Cf. A151799, A007917.

coQ[n_]:=PrimeQ[FromDigits[Flatten[IntegerDigits[{NextPrime[n,-1],n}]]]]; Select[Range[3,513],coQ[#]&] (* Jayanta Basu, May 30 2013 *)
Select[Range[2,550],PrimeQ[NextPrime[#,-1]*10^IntegerLength[#]+#]&] (* Harvey P. Dale, Nov 22 2020 *)

isok(n)=n>2 && isprime(fromdigits(concat(digits(precprime(n-1)), digits(n)))) \\ Andrew Howroyd, Aug 13 2024

from sympy import isprime, prevprime
def aupto(m):
  return [k for k in range(3, m+1) if isprime(int(str(prevprime(k))+str(k)))]
print(aupto(513)) # Michael S. Branicky, Mar 09 2021

Offset changed by Andrew Howroyd, Aug 13 2024

A034809 Numbers k such that the concatenation of k and previous_prime(k) is a prime.

Original entry on oeis.org

4, 5, 9, 10, 16, 24, 33, 36, 42, 46, 51, 53, 56, 59, 63, 66, 67, 69, 75, 76, 78, 81, 87, 96, 102, 106, 108, 111, 114, 116, 123, 125, 129, 130, 135, 137, 144, 145, 147, 148, 153, 156, 159, 170, 171, 177, 179, 180, 184, 187, 190, 192, 195, 196, 198, 207, 211, 214

Offset: 1

Patrick De Geest, Oct 15 1998

			k=156 is a term because the largest prime < 156 is 151 and '156151' is a prime.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harvey P. Dale)

Cf. A034808-A034821.

Select[Range[250],PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[ NextPrime[ #,-1]]]]]&] (* Harvey P. Dale, Jul 10 2017 *)

from sympy import isprime, prevprime
def ok(n): return isprime(int(str(n) + str(prevprime(n))))
print(list(filter(ok, range(3, 215)))) # Michael S. Branicky, Apr 05 2021

A034810 Concatenation of 'prevprime(k) and k and nextprime(k)' is a prime.

Original entry on oeis.org

7, 10, 13, 16, 19, 36, 37, 40, 43, 46, 58, 63, 74, 85, 88, 97, 98, 104, 125, 132, 143, 153, 156, 164, 168, 169, 175, 188, 196, 203, 206, 222, 224, 233, 241, 269, 292, 304, 305, 308, 311, 317, 331, 333, 338, 344, 359, 364, 367, 368, 372, 382, 389, 395, 397, 409

Offset: 1

Patrick De Geest, Oct 15 1998

			n=88 --> previous prime is 83, next prime is 89, thus '838889' is a prime.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A034808-A034821.

coQ[n_]:=PrimeQ[FromDigits[Flatten[IntegerDigits[{NextPrime[n,-1],n,NextPrime[n]}]]]]; Select[Range[3,409],coQ[#]&] (* Jayanta Basu, May 30 2013 *)

isok(n)=n>2 && isprime(fromdigits(concat([digits(precprime(n-1)), digits(n), digits(nextprime(n+1))]))) \\ Andrew Howroyd, Aug 13 2024

Offset changed by Andrew Howroyd, Aug 13 2024

A034814 Concatenations C1 and C2 are both prime (see the comment lines).

Original entry on oeis.org

9, 51, 63, 87, 111, 123, 153, 177, 207, 211, 239, 263, 273, 289, 327, 333, 343, 359, 407, 471, 489, 497, 513, 541, 597, 621, 651, 659, 663, 681, 687, 693, 697, 747, 753, 793, 819, 831, 869, 909, 977, 987, 1027, 1041, 1089, 1131, 1143, 1239, 1491, 1611

Offset: 1

Patrick De Geest, Oct 15 1998

C1 = 'prevprime(k) followed by k'.

C2 = 'k followed by prevprime(k)'.

			n=747 -> previous prime is 743, thus '743747' and '747743' are both primes.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A034808 and A034809.

Cf. A034810-A034821.

Offset changed by Andrew Howroyd, Aug 13 2024

A034811 Concatenation of 'nextprime(k) and k and prevprime(k)' is a prime.

Original entry on oeis.org

8, 23, 32, 36, 44, 54, 66, 74, 77, 84, 91, 104, 113, 115, 122, 130, 132, 162, 174, 178, 187, 188, 191, 204, 212, 222, 224, 232, 235, 259, 267, 281, 286, 295, 302, 305, 317, 325, 344, 353, 367, 368, 384, 389, 391, 401, 406, 427, 430, 433, 457, 458, 461, 464

Offset: 1

Patrick De Geest, Oct 15 1998

			n=353 -> next prime is 359, previous prime is 349, thus '359353349' is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034808-A034821.

Select[Range[500],PrimeQ[FromDigits[Flatten[IntegerDigits/@ { NextPrime[#], #, NextPrime[ #,-1]}]]]&] (* Harvey P. Dale, Jan 12 2016 *)

isok(n)=n>2 && isprime(fromdigits(concat([digits(nextprime(n+1)), digits(n), digits(precprime(n-1))]))) \\ Andrew Howroyd, Aug 13 2024

Offset changed by Andrew Howroyd, Aug 13 2024

A034812 Concatenation of 'prevprime(k) and k and prevprime(k)' is a prime.

Original entry on oeis.org

5, 8, 9, 12, 14, 17, 19, 23, 25, 28, 31, 33, 38, 39, 41, 42, 47, 48, 51, 60, 61, 62, 63, 69, 71, 72, 75, 77, 78, 80, 81, 84, 85, 91, 102, 104, 105, 111, 124, 126, 149, 150, 167, 181, 182, 189, 192, 194, 215, 222, 227, 230, 233, 242, 243, 256, 271, 273, 283, 288, 308

Offset: 1

Patrick De Geest, Oct 15 1998

			n=222 -> previous prime is 211, thus '211222211' is a prime.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A034808-A034821.

okQ[n_] := Module[{idppn=IntegerDigits[NextPrime[n,-1]]}, PrimeQ[FromDigits[Join[idppn, IntegerDigits[n], idppn]]]]; Select[Range[350], okQ] (* Harvey P. Dale, Jan 07 2011 *)

isok(n)=my(x=digits(precprime(n-1))); n>2 && isprime(fromdigits(concat([x, digits(n), x]))) \\ Andrew Howroyd, Aug 13 2024

Offset changed by Andrew Howroyd, Aug 13 2024

A034813 Concatenations C1 and C2 are both prime (see the comment lines).

Original entry on oeis.org

36, 74, 104, 132, 188, 222, 224, 305, 317, 344, 367, 368, 389, 457, 458, 475, 540, 572, 584, 608, 631, 676, 682, 689, 697, 738, 756, 760, 781, 797, 829, 841, 893, 910, 911, 914, 928, 982, 1018, 1104, 1122, 1178, 1186, 1317, 1328, 1391, 1402, 1406, 1518

Offset: 0

Patrick De Geest, Oct 15 1998

C1 = 'prevprime(n) followed by n followed by nextprime(n)'

C2 = 'nextprime(n) followed by n followed by prevprime(n)'

			n=797 -> previous prime is 787, next prime is 809, thus '787797809' and '809797787' are both primes.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A034808-A034821.

c1c2Q[n_]:=Module[{ia=IntegerDigits[NextPrime[n,-1]],ib=IntegerDigits[n], ic= IntegerDigits[NextPrime[n]]}, AllTrue[{FromDigits[Join[ia,ib,ic]], FromDigits[Join[ic,ib,ia]]},PrimeQ]]; Select[Range[1600],c1c2Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 06 2018 *)

A034820 Concatenations C1 and C2 and C3 and C4 are all prime (see the comment lines).

Original entry on oeis.org

16776, 41719, 164612, 188435, 188682, 312184, 317594, 392771, 397617, 450413, 476055, 486283, 492240, 497913, 539471, 584029, 620029, 640883, 648445, 656757, 903009, 992790, 993475, 995917, 1045387, 1082078, 1194606, 1252496, 1322841

Offset: 0

Patrick De Geest, Oct 15 1998

C1 = 'prevprime(n) followed by n followed by prevprime(n)'
C2 = 'prevprime(n) followed by n followed by nextprime(n)'
C3 = 'nextprime(n) followed by n followed by prevprime(n)'
C4 = 'nextprime(n) followed by n followed by nextprime(n)'

			n=41719 -> next prime is 41729, previous prime is 41687, thus '416874171941687' and '416874171941729' and '417294171941687' and '417294171941729' are all four primes.

Cf. A034808-A034821.

A034815 Concatenations C1 and C2 are both prime (see the comment lines).

Original entry on oeis.org

33, 51, 53, 63, 111, 123, 129, 211, 237, 273, 357, 489, 519, 573, 597, 609, 639, 651, 653, 657, 669, 681, 687, 747, 753, 819, 831, 873, 891, 987, 997, 1071, 1611, 1881, 2037, 2049, 2247, 2271, 2613, 2763, 3063, 3267, 3393, 3573, 3969, 4251, 4263, 4293

Offset: 0

Patrick De Geest, Oct 15 1998

C1 = 'nextprime(n) followed by n'

C2 = 'n followed by prevprime(n)'

			n=1881 -> next prime is 1889, previous prime is 1879, thus '18891881' and '18811879' are both primes.

Cf. A034808-A034821.

okQ[n_]:=Module[{idn=IntegerDigits[n],ida,idb},ida=IntegerDigits[NextPrime[n,-1]];idb=IntegerDigits[NextPrime[n]];PrimeQ[FromDigits[Join[idn,ida]]]&&PrimeQ[FromDigits[Join[idb,idn]]]]
Select[Range[5000],okQ]  (* Harvey P. Dale, Dec 25 2010 *)

A034816 Concatenations C1 and C2 are both prime (see the comment lines).

Original entry on oeis.org

9, 51, 63, 117, 123, 157, 183, 213, 217, 219, 257, 263, 321, 327, 333, 423, 441, 473, 541, 597, 621, 687, 693, 723, 777, 879, 909, 987, 1101, 1143, 1187, 1387, 1459, 1551, 1887, 2127, 2179, 2193, 2817, 3197, 3339, 3513, 3633, 4029, 4209, 4429, 4529, 4557

Offset: 0

Patrick De Geest, Oct 15 1998

C1 = 'prevprime(n) followed by n'

C2 = 'n followed by nextprime(n)'

			n=1551 -> previous prime is 1549, next prime is 1553, thus '15491551' and '15511553' are both primes.

Cf. A034808-A034821.

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A034808 Concatenation of 'prevprime(k) and k' is a prime.

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A034809 Numbers k such that the concatenation of k and previous_prime(k) is a prime.

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A034810 Concatenation of 'prevprime(k) and k and nextprime(k)' is a prime.

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A034814 Concatenations C1 and C2 are both prime (see the comment lines).

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A034811 Concatenation of 'nextprime(k) and k and prevprime(k)' is a prime.

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A034812 Concatenation of 'prevprime(k) and k and prevprime(k)' is a prime.

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Mathematica

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A034813 Concatenations C1 and C2 are both prime (see the comment lines).

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Programs

Mathematica

A034820 Concatenations C1 and C2 and C3 and C4 are all prime (see the comment lines).

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