cp's OEIS Frontend

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

%I A076653 #29 Dec 05 2015 17:25:53
%S A076653 2,23,3,31,11,13,37,7,71,17,73,307,79,97,701,19,907,709,911,101,103,
%T A076653 311,107,719,919,929,937,727,733,313,317,739,941,109,947,743,331,113,
%U A076653 337,751,127,757,761,131,137,769,953,347,773,349,967,787,797,7001,139,971
%N A076653 Smallest prime number not occurring earlier and starting with the final digit of the previous term.
%C A076653 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
%C A076653 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
%C A076653 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
%C A076653 From _Danny Rorabaugh_, Dec 01 2015: (Start)
%C A076653 We can initiate with a different prime p (see the a-file):
%C A076653     p=3: [a(3), a(4), ...];
%C A076653     p=5: [5, 53, a(3), a(4), ...];
%C A076653     p=7: [7, 71, 11, 13, 3, 31, 17, 73, 37, ...];
%C A076653 etc.
%C A076653 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:
%C A076653     Starting with 1, or starting with 2/4/5/6/8 and ending with 1
%C A076653   [11, 13, 17, 19, 41, 61, 101, 103, 107, 109, 113, 127, 131, 137, ...];
%C A076653     Starting with 3, or starting with 2/4/5/6/8 and ending with 2/3/5
%C A076653   [2, 3, 5, 23, 31, 37, 43, 53, 83, 223, 233, 263, 283, 293, 307, ...];
%C A076653     Starting with 7, or starting with 2/4/5/6/8 and ending with 7
%C A076653   [7, 47, 67, 71, 73, 79, 227, 257, 277, 457, 467, 487, 547, 557, ...];
%C A076653     Starting with 9, or starting with 2/4/5/6/8 and ending with 9
%C A076653   [29, 59, 89, 97, 229, 239, 269, 409, 419, 439, 449, 479, 499, ...].
%C A076653 (End)
%H A076653 Zak Seidov, <a href="/A076653/b076653.txt">Table of n, a (n) for n = 1..10000</a>
%H A076653 Danny Rorabaugh, <a href="/A076653/a076653.txt">A076653-variants with prime initial values 2<=a(0)<=997</a>
%p A076653 N:= 10^5: # get all terms before the first a(n) > N
%p A076653 Primes:= select(isprime,[seq(i,i=3..N,2)]):
%p A076653 Inits:= map(p -> floor(p/10^ilog10(p)), Primes):
%p A076653 for d in [1,2,3,7,9] do
%p A076653   Id[d]:= select(t -> Inits[t]=d, [$1..nops(Inits)]); p[d]:= 1;w[d]:= nops(Id[d]);
%p A076653 od:
%p A076653 A[1]:= 2:
%p A076653 for n from 2 do
%p A076653   d:= A[n-1] mod 10;
%p A076653   if p[d] > w[d] then break fi;
%p A076653   A[n]:= Primes[Id[d][p[d]]];
%p A076653   p[d]:= p[d]+1;
%p A076653 od:
%p A076653 seq(A[i],i=1..n-1); # _Robert Israel_, Dec 01 2015
%t A076653 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_ *)
%t A076653 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 *)
%o A076653 (Sage)
%o A076653 def A076653(lim,p=2):
%o A076653     A = [p]
%o A076653     while len(A)<lim:
%o A076653         for q in Primes():
%o A076653             if (q not in A) and (str(A[-1])[-1]==str(q)[0]):
%o A076653                 A.append(q)
%o A076653                 break
%o A076653     return A
%o A076653 A076653(56) # _Danny Rorabaugh_, Dec 01 2015
%Y A076653 Cf. A076652, A076654, A082238, A089755, A107809, A180022.
%K A076653 base,nonn,look
%O A076653 1,1
%A A076653 _Amarnath Murthy_, Oct 28 2002
%E A076653 More terms from _Robert G. Wilson v_, Nov 17 2005