A180022 -id:A180022 - 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

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],", ") )

A321523 List of pairs: primes whose reversal is also prime, each followed by its reversal.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 13, 31, 17, 71, 31, 13, 37, 73, 71, 17, 73, 37, 79, 97, 97, 79, 101, 101, 107, 701, 113, 311, 131, 131, 149, 941, 151, 151, 157, 751, 167, 761, 179, 971, 181, 181, 191, 191, 199, 991, 311, 113, 313, 313, 337, 733, 347, 743, 353, 353

Offset: 1

Kritsada Moomuang, Nov 12 2018

			The sequence begins:
     2,  2;
     3,  3;
     5,  5;
     7,  7;
    11, 11;
    13, 31;
    17, 71;
    31, 13;
    37, 73;
    71, 17;
...
107 has its reversal as 701.
971 has its reversal as 179.

Subsequence of A135020.

Cf. A000040, A004086, A007500, A095180, A180022.

Flatten@ Table[ If[PrimeQ[r = IntegerReverse@ p], {p,r}, {}], {p, Prime@ Range@ 71}] (* Giovanni Resta, Nov 13 2018 *)

forprime(p=1, 353, r=fromdigits(Vecrev(digits(p))); if (isprime(r), print1(p ", " r ", "))) \\ Rémy Sigrist, Nov 16 2018

a(2n-1) = A007500(n).
a(2n) = A004086(A007500(n)).
a(2n) = A095180(n). - Rémy Sigrist, Nov 16 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A321523 List of pairs: primes whose reversal is also prime, each followed by its reversal.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula