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

Author

Amarnath Murthy, Oct 28 2002

Keywords

• base
• nonn
• look

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

Crossrefs

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

Programs

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 *)

Sage

def A076653(lim,p=2):
    A = [p]
    while len(A)A076653(56) # Danny Rorabaugh, Dec 01 2015

Extensions

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