A089755 -id:A089755 - cp's OEIS frontend

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

Amarnath Murthy, Oct 28 2002

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)

Zak Seidov, Table of n, a (n) for n = 1..10000
Danny Rorabaugh, A076653-variants with prime initial values 2<=a(0)<=997

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

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

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 *)

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

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

A262283 a(1)=2. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 31, 17, 71, 19, 97, 73, 37, 79, 907, 701, 101, 103, 307, 709, 911, 113, 131, 311, 1103, 1031, 313, 137, 373, 733, 331, 317, 173, 739, 397, 971, 719, 191, 919, 193, 937, 379, 797, 977, 773, 7307, 3079, 7901, 9011, 1109, 109, 929, 29, 941, 41

Offset: 1

N. J. A. Sloane, Sep 18 2015

If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
Leading zeros in s are ignored.
The sequence is infinite, since there infinitely many primes that start with s (see the comments in A080165).
The data in the b-file suggests that there are infinitely many primes that do not appear. Hoever, at present that is no proof that even one prime (23, say) never appears. - N. J. A. Sloane, Sep 20 2015
Alois P. Heinz points out that a(n) = A262282(n+29) starting at the 103rd term. - N. J. A. Sloane, Sep 19 2015

			a(1)=2, so s is the empty string, so a(2) is the smallest missing prime, 3. After a(6)=13, s=3, so a(7) is the smallest missing prime that starts with 3, which is 31.

Alois P. Heinz, Table of n, a(n) for n = 1..676

Cf. A080165, A089755, A262282, A262350.

import Data.List (isPrefixOf, delete)
a262283 n = a262283_list !! (n-1)
a262283_list = 2 : f "" (map show $ tail a000040_list) where
   f xs pss = (read ys :: Integer) :
              f (dropWhile (== '0') ys') (delete ys pss)
              where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
-- Reinhard Zumkeller, Sep 19 2015

More terms from Alois P. Heinz, Sep 18 2015

A262282 a(1)=11. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

Original entry on oeis.org

11, 13, 3, 2, 5, 7, 17, 71, 19, 97, 73, 31, 101, 103, 37, 79, 907, 701, 107, 709, 911, 113, 131, 311, 1103, 1031, 313, 137, 373, 733, 331, 317, 173, 739, 397, 971, 719, 191, 919, 193, 937, 379, 797, 977, 773, 7307, 307, 727, 271, 7103, 1033, 337, 3701, 7013

Offset: 1

Franklin T. Adams-Watters and N. J. A. Sloane, Sep 18 2015

If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
Leading zeros in s are ignored.
The b-file suggests that there are infinitely many primes that do not appear in the sequence. However, there is no proof at present that any particular prime (23, say) never appears.
Alois P. Heinz points out that this sequence and A262283 eventually merge (see the latter entry for details). - N. J. A. Sloane, Sep 19 2015
A variant without the prime number condition: A262356. - Reinhard Zumkeller, Sep 19 2015

			a(1)=11, so s=1, a(2) is smallest missing prime that starts with 1, so a(2)=13. Then s=3, so a(3)=3. Then s is the empty string, so a(4)=2, and so on.

Alois P. Heinz, Table of n, a(n) for n = 1..705

Suggested by A089755. Cf. A262283.

Cf. A262356.

import Data.List (isPrefixOf, delete)
a262282 n = a262282_list !! (n-1)
a262282_list = 11 : f "1" (map show (delete 11 a000040_list)) where
   f xs pss = (read ys :: Integer) :
              f (dropWhile (== '0') ys') (delete ys pss)
              where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
-- Reinhard Zumkeller, Sep 19 2015

More terms from Alois P. Heinz, Sep 18 2015

A262254 a(1) = 11; for n>1, a(n) is the smallest prime that starts with the least significant digit of the previous term and has not occurred earlier.

Original entry on oeis.org

11, 13, 31, 17, 71, 19, 97, 73, 37, 79, 907, 701, 101, 103, 307, 709, 911, 107, 719, 919, 929, 937, 727, 733, 311, 109, 941, 113, 313, 317, 739, 947, 743, 331, 127, 751, 131, 137, 757, 761, 139, 953, 337, 769, 967, 773, 347, 787, 797, 7001, 149, 971, 151, 157, 7013, 349, 977, 7019, 983

Offset: 1

Maghraoui Abdelkader, Sep 16 2015

This sequence is different from A089755, as this one does not include single-digit primes.
This sequence is different from A089755, for which a(n+1) uses all the digit of a(n) except the most-significant digit of a(n).
In this sequence, a(n+1) uses only the least-significant digit of a(n).

			a(3) = 31 where the most significant digit of 31 is 3, which is the least significant digit of a(2) = 13.

Maghraoui Abdelkader, Table of n, a(n) for n = 1..200

Cf. A076653, A082238, A089755, A107809, A180022.

f[s_List] := Block[{lsd = Mod[s[[-1]], 10], p = 3}, While[ IntegerDigits[p][[1]] != lsd || MemberQ[s, p], p = NextPrime@ p]; Append[s, p]]; s = {11}; s = Nest[f, s, 70] (* Robert G. Wilson v, Sep 16 2015 *)

msd(n)=(n\10^(#Str(n)-1)); l1=1;l3=1;l7=1;l9=1; q=1; lsd(n)=n%10;
t1 = vector(200); i=1;forprime(n=11,10000, if( digits(n)[1]==1,t1[i]=n; i=i+1 ; )) ;
t3 = vector(130); i=1;forprime(n=31,3900 ,  if( digits(n)[1]==3,t3[i]=n; i=i+1 ; ));
t7 = vector(130); i=1;forprime(n=71,70000, if( digits(n)[1]==7,t7[i]=n; i=i+1 ; )) ;
t9 = vector(130); i=1;forprime(n=97,90000, if( digits(n)[1]==9,t9[i]=n; i=i+1 ; )) ;lsd(n)=n%10;
t = vector(200);
findnextnumber(m) =
{if(lsd(m)==1, l1=t1[i1] ; i1=i1+1;return(l1);); if(lsd(m)==3, l3=t3[i3];  i3=i3+1 ; return(l3););
if(lsd(m)==7, l7=t7[i7] ; i7=i7+1;return(l7);); if(lsd(m)==9, l9=t9[i9];  i9=i9+1;return(l9);); }
i1=1;  i3=1;  i7=1;  i9=1;  j=1;s=11;   for(n=1,200,   p=findnextnumber(s) ; t[j]=p; s=p;j++);for(i=1,200, print1(t[i],", ") )

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

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A262283 a(1)=2. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

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A262282 a(1)=11. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

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A262254 a(1) = 11; for n>1, a(n) is the smallest prime that starts with the least significant digit of the previous term and has not occurred earlier.

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