A076653 Smallest prime number not occurring earlier and starting with the final digit of the previous term.

2, 23, 3, 31, 11, 13, 37, 7, 71, 17, 73, 307, 79, 97, 701, 19, 907, 709, 911, 101, 103, 311, 107, 719, 919, 929, 937, 727, 733, 313, 317, 739, 941, 109, 947, 743, 331, 113, 337, 751, 127, 757, 761, 131, 137, 769, 953, 347, 773, 349, 967, 787, 797, 7001, 139, 971

Offset

1,1

Comments

This sequence is infinite but still does not contain all the primes. There is no way for 5 to appear, nor any higher prime starting with 5. - Alonso del Arte, Sep 19 2015

Moreover, it is an obvious fact that there is no way for any prime starting with 2 (aside from the first two), 4, 6 or 8 to appear. - Altug Alkan, Sep 20 2015

Apart from the first two terms, this sequence is identical to how it would be if it were to start with 5 and 53 instead of 2 and 23. - Maghraoui Abdelkader, Sep 22 2015

From Danny Rorabaugh, Dec 01 2015: (Start)

We can initiate with a different prime p (see the a-file):

p=3: [a(3), a(4), ...];

p=5: [5, 53, a(3), a(4), ...];

p=7: [7, 71, 11, 13, 3, 31, 17, 73, 37, ...];

etc.

Define p~q to mean that the sequences generated by p and q eventually coincide (with different offset allowed). For example, we can see that 2~3~5, but it appears these are not equivalent to 7. Empirically, there are exactly four equivalence classes of primes:

Starting with 1, or starting with 2/4/5/6/8 and ending with 1

[11, 13, 17, 19, 41, 61, 101, 103, 107, 109, 113, 127, 131, 137, ...];

Starting with 3, or starting with 2/4/5/6/8 and ending with 2/3/5

[2, 3, 5, 23, 31, 37, 43, 53, 83, 223, 233, 263, 283, 293, 307, ...];

Starting with 7, or starting with 2/4/5/6/8 and ending with 7

[7, 47, 67, 71, 73, 79, 227, 257, 277, 457, 467, 487, 547, 557, ...];

Starting with 9, or starting with 2/4/5/6/8 and ending with 9

[29, 59, 89, 97, 229, 239, 269, 409, 419, 439, 449, 479, 499, ...].

(End)

Links

Zak Seidov, Table of n, a (n) for n = 1..10000

Danny Rorabaugh, A076653-variants with prime initial values 2<=a(0)<=997

Maple

N:= 10^5: # get all terms before the first a(n) > N
Primes:= select(isprime, [seq(i, i=3..N, 2)]):
Inits:= map(p -> floor(p/10^ilog10(p)), Primes):
for d in [1, 2, 3, 7, 9] do
  Id[d]:= select(t -> Inits[t]=d, [$1..nops(Inits)]); p[d]:= 1; w[d]:= nops(Id[d]);
od:
A[1]:= 2:
for n from 2 do
  d:= A[n-1] mod 10;
  if p[d] > w[d] then break fi;
  A[n]:= Primes[Id[d][p[d]]];
  p[d]:= p[d]+1;
od:
seq(A[i], i=1..n-1); # Robert Israel, Dec 01 2015

Mathematica

prevLastDigPrime[seq_] := Block[{k = 1, lastDigit = Mod[Last@seq, 10]}, While[p = Prime@k; MemberQ[seq, p] || lastDigit != Quotient[p, 10^Floor[Log[10, p]]], k++]; Append[seq, p]]; Nest[prevLastDigPrime, {2}, 55] (* Robert G. Wilson v *)
A076653 = {2}; Do[k = 2; d = Last@IntegerDigits@A076653[[n - 1]]; While[Or[MemberQ[A076653, k], First@IntegerDigits@k != d], k = NextPrime@k]; AppendTo[A076653, k], {n, 2, 60}]; A076653 (* Michael De Vlieger, Sep 21 2015 *)

Prog

(Sage)
def A076653(lim, p=2):
    A = [p]
    while len(A)<lim:
        for q in Primes():
            if (q not in A) and (str(A[-1])[-1]==str(q)[0]):
                A.append(q)
                break
    return A
A076653(56) # Danny Rorabaugh, Dec 01 2015

Crossrefs

Cf. A076652, A076654, A082238, A089755, A107809, A180022.

Sequence in context: A107801 A262702 A360534 * A114008 A110354 A245628

Adjacent sequences: A076650 A076651 A076652 * A076654 A076655 A076656

Keyword

base,nonn,look

Author

Amarnath Murthy, Oct 28 2002

Extensions

More terms from Robert G. Wilson v, Nov 17 2005

Status

approved