A083569 -id:A083569 - cp's OEIS frontend

A084721 Primes arising in A083569.

2, 5, 5, 11, 11, 11, 11, 17, 17, 23, 23, 23, 23, 29, 29, 37, 37, 37, 37, 37, 37, 47, 47, 47, 47, 53, 53, 59, 59, 59, 59, 67, 67, 67, 67, 73, 73, 79, 79, 79, 79, 89, 89, 89, 89, 89, 89, 97, 97, 101, 101, 107, 107, 107, 107, 113, 113, 127, 127, 127, 127, 127, 127, 127, 127

Offset: 1

Amarnath Murthy, Jun 12 2003

nonn

More precisely, a(n)=n+A083569(n). The sequence is strictly increasing, by construction/definition of A083569. - M. F. Hasler, Apr 11 2015

Cf. A083569.

More terms from David Wasserman, Jan 03 2005

A071065 A permutation of natural numbers: a(1)=2 a(n) is the smallest integer different from a(1), a(2), a(3), ..., a(n-1) such that a(n)+n is prime.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 10, 9, 8, 7, 12, 11, 16, 15, 14, 13, 20, 19, 18, 17, 22, 21, 24, 23, 28, 27, 26, 25, 30, 29, 36, 35, 34, 33, 32, 31, 42, 41, 40, 39, 38, 37, 46, 45, 44, 43, 50, 49, 48, 47, 52, 51, 54, 53, 58, 57, 56, 55, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 70, 69, 78, 77

Offset: 1

Benoit Cloitre, May 26 2002

a(a(n))=n: a self-inverse permutation with A010051(a(n)+n)=1. - Reinhard Zumkeller, Nov 25 2004

Index entries for sequences that are permutations of the natural numbers

Cf. A010051, A083569.

Corrected and extended by Lior Manor, May 27 2002

A359973 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the concatenation of the decimal digits of n and a(n) or of a(n) and n yields a prime number.

Original entry on oeis.org

1, 3, 2, 7, 9, 13, 4, 11, 5, 19, 8, 17, 6, 23, 31, 21, 12, 47, 10, 27, 16, 37, 14, 41, 39, 33, 20, 43, 32, 49, 15, 29, 26, 57, 59, 71, 22, 51, 25, 73, 24, 53, 28, 63, 61, 79, 18, 77, 30, 81, 38, 97, 42, 83, 69, 89, 34, 67, 35, 91, 45, 87, 44, 109, 99, 103, 58

Offset: 1

Rémy Sigrist, Jan 20 2023

Leading zeros are ignored.
This sequence is a self-inverse permutation of the positive integers (for any positive number v, there are infinitely many prime numbers starting with 10*v+1, so infinitely many prime numbers that are the concatenation of v and some other positive integer).
There is only one fixed point: a(1) = 1.

			The first terms, alongside the corresponding prime numbers, are:
  n   a(n)  Corresponding prime numbers
  --  ----  ---------------------------
   1     1  {11}
   2     3  {23}
   3     2  {23}
   4     7  {47}
   5     9  {59}
   6    13  {613}
   7     4  {47}
   8    11  {811}
   9     5  {59}
  10    19  {1019}
  11     8  {811}
  12    17  {1217}

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing a(n) coprime to 10 in red and other numbers in dark blue.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in dark green, and other numbers in blue, showing powerful numbers that are not prime powers in light blue.
Mathematics StackExchange, Proof that there are infinitely many prime numbers starting with a given digit string
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A083569, A088629.

nn = 120; c[] := False; a[1] = 1; c[1] = True; u = 2; Q[n] := AnyTrue[{FromDigits[Join[d, #]], FromDigits[Join[#, d]]} & @@ {IntegerDigits[n], d}, PrimeQ]; Do[Set[{k, d}, {u, IntegerDigits[n]}]; While[Nand[! c[k], Q[k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jan 21 2023 *)

PARI
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See Links section.
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A256864 Primes not occurring in A084721.

Original entry on oeis.org

3, 7, 13, 19, 31, 41, 43, 61, 71, 83, 103, 109, 131, 137, 139, 151, 157, 167, 179, 193, 197, 199, 223, 229, 241, 257, 271, 281, 283, 311, 337, 349, 383, 421, 433, 439, 461, 463, 487, 503, 523, 547, 571

Offset: 1

M. F. Hasler, Apr 11 2015

nonn

Complement of the range of A084721 within the primes A000040.

Cf. A083569, A084721.

N=300; setminus(primes(primepi(A083569(N)+N)),Set(vector(N,n,A083569(n)+n))) \\ M. F. Hasler, Apr 11 2015

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A084721 Primes arising in A083569.

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A071065 A permutation of natural numbers: a(1)=2 a(n) is the smallest integer different from a(1), a(2), a(3), ..., a(n-1) such that a(n)+n is prime.

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A359973 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the concatenation of the decimal digits of n and a(n) or of a(n) and n yields a prime number.

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A256864 Primes not occurring in A084721.

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PARI