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A076653 Smallest prime number not occurring earlier and starting with the final digit of the previous term. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A323396 A107801 A262702 * 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

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Last modified December 9 18:21 EST 2022. Contains 358703 sequences. (Running on oeis4.)