A094044 -id:A094044 - cp's OEIS frontend

A094045 Alternate composite and prime numbers not included earlier such that every concatenation of a pair of terms is a prime: a(2n) is prime and a(2n-1) is nonprime.

1, 3, 49, 19, 9, 7, 27, 11, 51, 13, 21, 29, 39, 17, 33, 23, 57, 37, 69, 47, 87, 31, 63, 43, 91, 61, 99, 41, 77, 53, 81, 67, 93, 71, 119, 59, 123, 73, 121, 97, 117, 79, 111, 103, 141, 101, 159, 113, 143, 89, 153, 83, 177, 109, 133, 157, 189, 127, 207, 139, 169, 151, 171, 131

Offset: 1

Robert G. Wilson v, Apr 23 2004

Conjecture: 2 and 5 are the only two nonmembers.

			a(3)=49 => 349 is a prime but not necessarily 1349, which by the way it
is not.

Cf. A094043, A094044.

p = Prime[ Range[ 500]]; np = Drop[ Complement[ Range[ 500], p], 1]; a[1] = 1; a[n_] := a[n] = Block[{k = 1, q = IntegerDigits[a[n - 1]]}, If[ EvenQ[n], While[ !PrimeQ[ FromDigits[ Join[q, IntegerDigits[ p[[k]] ]]]], k++ ]; q = p[[k]]; p = Delete[p, k]; q, While[ !PrimeQ[ FromDigits[ Join[q, IntegerDigits[ np[[k]] ]]]], k++ ]; q = np[[k]]; np = Delete[np, k]; q]]; Table[ a[n], {n, 60}]

A345966 The succession of nonprime and prime terms is kept when you consider the sequence formed by the successive sums a(n) + a(n+1). This is the lexicographically earliest sequence of distinct positive terms with this property.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 8, 7, 10, 11, 12, 9, 13, 16, 14, 18, 15, 17, 20, 19, 22, 23, 24, 21, 25, 26, 28, 27, 29, 30, 32, 31, 36, 33, 35, 34, 38, 37, 42, 39, 41, 48, 40, 44, 43, 46, 45, 47, 50, 49, 51, 53, 54, 52, 56, 55, 57, 58, 59, 68, 60, 61, 66, 62, 63, 65, 64, 69, 67, 70, 71, 78, 72, 73, 76, 74, 79, 84, 75, 77

Offset: 1

Eric Angelini and Carole Dubois, Jun 30 2021

nonn

Here is the succession of nonprimes and primes in the sequence:
1, 3, 2, 5, 6, 4, 8, 7, 10, 11, 12, 9, 13, 16, 14, 18, 15,
n  p  p  p  n  n  n  p   n   p   n  n   p   n   n   n   n
The same succession is formed by a(n) + a(n+1):
4, 5, 7, 11, 10, 12, 15, 17, 21, 23, 21, 22, 29, 30, 32, 33, 32
n  p  p   p   n   n   n   p   n   p   n   n   p   n   n   n   n

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A094044, A345903.

seq[n_] := Module[{s = {1}, q, k}, Do[q = PrimeQ[s[[-1]]]; k = 1; While[!FreeQ[s, k] || PrimeQ[s[[-1]] + k] != q, k++]; AppendTo[s, k], {n}]; s]; seq[100] (* Amiram Eldar, Jun 30 2021 *)

cp's OEIS Frontend

A094045 Alternate composite and prime numbers not included earlier such that every concatenation of a pair of terms is a prime: a(2n) is prime and a(2n-1) is nonprime.

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