A034593 -id:A034593 - cp's OEIS frontend

A034591 Numbers whose concatenation with the next prime yields a prime.

0, 2, 6, 8, 9, 12, 27, 31, 36, 42, 44, 45, 51, 56, 63, 68, 69, 78, 83, 86, 93, 94, 96, 98, 102, 104, 105, 108, 115, 117, 118, 123, 126, 132, 138, 150, 151, 154, 156, 157, 160, 167, 176, 180, 183, 186, 192, 194, 195, 199, 201, 205, 213, 217, 219, 225, 230, 233

Offset: 1

Patrick De Geest, Oct 15 1998

			8 is a term because nextprime(8) = 11 and 811 is prime.

See also A034592, A034593, A034594, A034596, A030459.

Select[Range[0,250],PrimeQ@FromDigits@Flatten[IntegerDigits/@{#,NextPrime@#}]&] (* Giorgos Kalogeropoulos, Jul 30 2021 *)

isok(m) = isprime(eval(Str(m, nextprime(m+1)))); \\ Michel Marcus, Oct 03 2014 and Jul 30 2021

A036341 Concatenation of prime p and nextprime(p) is prime -> cycles of 4 steps possible.

Original entry on oeis.org

127787377, 1510818931, 3147482977, 3307903909, 3621408103, 3964636549, 7224979183, 7881628361, 8043354509, 8740618759, 10091655397, 10164789521, 10194261067, 10380609119, 10436432219, 11472028801, 12059694769, 13887362681, 14880879703, 15197340049

Offset: 1

Patrick De Geest, Dec 15 1998

a(1)-a(5) found by Jo Yeong Uk (hyukjo(AT)sigma.chungnam.ac.kr).

Giovanni Resta, Table of n, a(n) for n = 1..4000
Carlos Rivera, Puzzle 29. P_i = P_(i-1) & nxtprm(P_(i-1)), P_i = prime for i => 1, The Prime Puzzles and Problems Connection.

Cf. A030459, A034593, A036339, A036340.

Offset corrected and terms a(6) and beyond added by Giovanni Resta, May 14 2020

A036340 Concatenation of prime p and nextprime(p) is prime -> cycles of 3 steps possible.

Original entry on oeis.org

467, 941, 959941, 3396199, 4858943, 5696101, 6475643, 7566133, 7584253, 7592261, 9305281, 9463877, 11430491, 13442243, 14374837, 15941473, 17414497, 17691997, 19584223, 21421849, 22310159, 22808459, 27601163, 29198881

Offset: 0

Patrick De Geest, Dec 15 1998

Terms from 3396199 up to 17691997 found by Jo Yeong Uk (hyukjo(AT)sigma.chungnam.ac.kr).

Carlos Rivera, Puzzle 29. Pi = P i-1&nxtprm(P i-1), Pi = prime for i => 1, The Prime Puzzles & Problems Connection.

Cf. A030459, A034593, A036339, A036341.

A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

Original entry on oeis.org

3, 2, 13681, 467, 127787377, 200603842261

Offset: 0

Giovanni Resta, May 14 2020

a(6) > 10^13.

			Let "|" denote concatenation.
3 | 5 = 35, which is not prime, so a(0) = 3.
2 | 3 = 23 (prime), 23 | 29 = 2329 (composite), so a(1) = 2.
13681 | 13687 (prime), 1368113687 | 1368113699 (prime), 13681136871368113699 | 13681136871368113711 (composite), so a(2) = 13681.

Carlos Rivera, Puzzle 29. P_i = P_(i-1) & nxtprm(P_(i-1)), P_i = prime for i => 1, The Prime Puzzles and Problems Connection.

Cf. A030459, A034593, A036339, A036340, A036341.

a[n_] := Block[{pp=1, p, q, c=-1}, While[ c!=n, c=0; p = pp = NextPrime@ pp; While[ PrimeQ[ q = FromDigits[ Join @@ IntegerDigits@{p, NextPrime@ p}]], c++; p = q]]; pp]; a /@ Range[0, 3]

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A034591 Numbers whose concatenation with the next prime yields a prime.

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A036341 Concatenation of prime p and nextprime(p) is prime -> cycles of 4 steps possible.

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A036340 Concatenation of prime p and nextprime(p) is prime -> cycles of 3 steps possible.

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A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

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