A075609 -id:A075609 - cp's OEIS frontend

A075607 a(1) = 1, a(n) = smallest number not occurring earlier such that the concatenation a(n-1) and a(n) is a prime.

1, 3, 7, 9, 11, 17, 21, 13, 19, 31, 37, 27, 29, 39, 23, 33, 43, 49, 51, 47, 59, 53, 81, 61, 63, 67, 79, 93, 41, 57, 83, 69, 71, 77, 89, 99, 73, 121, 97, 87, 103, 91, 127, 123, 113, 111, 109, 133, 117, 101, 107, 119, 129, 169, 151, 141, 131, 143, 137, 147, 139

Offset: 1

Amarnath Murthy, Sep 28 2002

Almost certainly a permutation of A045572. - David W. Wilson, Jan 15 2005

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A075608, A075609, A075610.

a = {{1}}; Do[k = 2; While[Nand[! MemberQ[a, #], PrimeQ@ FromDigits@ Join[a[[n - 1]], #]] &@ Set[d, IntegerDigits@ k], k++]; AppendTo[a, d], {n, 2, 61}]; FromDigits /@ a (* Michael De Vlieger, May 08 2017 *)

A075607(n,show=0,a=1,u=[])={for(n=2,n,show&&print1(a","); u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+2,u=u[2..-1]); forstep(k=u[1]+2, 9e9, 2, setsearch(u,k)&&next; isprime(eval(Str(a,k))) && (a=k) && break)); a} \\ Use 2nd, 3rd or 4th optional arg to print intermediate terms, to use another starting value or to exclude some terms. - M. F. Hasler, Nov 25 2015

More terms from Michel ten Voorde Jun 23 2003

A073641 a(1) = 2; a(n) = smallest prime not included earlier such that concatenation of two successive terms is a prime.

Original entry on oeis.org

2, 3, 7, 19, 13, 61, 31, 37, 67, 79, 103, 43, 73, 127, 139, 97, 151, 157, 109, 199, 181, 193, 163, 211, 229, 223, 241, 271, 277, 331, 283, 397, 337, 313, 307, 367, 457, 421, 349, 373, 379, 433, 439, 409, 463, 523, 487, 601, 541, 547, 499, 571, 673, 613, 577

Offset: 1

Amarnath Murthy, Aug 09 2002

Conjecture: every prime besides 5 is in this list. - Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003
It appears that the terms belong to A007645. There are no primes of form 6k-1 in this sequence. - Alexander Adamchuk, Aug 15 2006
The above conjecture by Cunningham (Apr 11 2003) is false: Since a(2)=3 and a(3)=7 == 1 mod 6, all subsequent terms must also be 1 mod 6 because concatenations of numbers 1 mod 6 with 5 mod 6 are 0 mod 3. - Bob Selcoe, Aug 25 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A075609.

N:= 10000: # to get all terms before the first one > N
A[1]:= 2:
Primes:= Vector(select(isprime,[seq(2*i+1 , i=1..floor((N-1)/2))])):
Nprimes:= LinearAlgebra:-Dimension(Primes):
Next:= Vector(Nprimes):
Prev:= Vector(Nprimes):
for i from 1 to Nprimes-1 do Next[i]:= i+1; Prev[i+1]:= i od:
first:= 1:
found:= true:
for n from 2 while found do
  i:= first;
  found:= false;
  while i <> 0 do
    p:= Primes[i];
    if isprime(10^(1+ilog10(p))*A[n-1] + p) then
      found:= true;
      A[n]:= p;
      if i = first then first:= Next[first]
      else Next[Prev[i]]:= Next[i]
      fi;
      if Next[i] <> 0 then
        Prev[Next[i]]:= Prev[i]
      fi;
      break
    fi;
    i:= Next[i];
  od
od:
seq(A[i],i=1..n-2); # Robert Israel, Aug 25 2015

t = {2}; Do[i = 2; While[! PrimeQ[FromDigits[Flatten[IntegerDigits[{Last[t], x = Prime[i]}]]]] || MemberQ[t, x], i++]; AppendTo[t, x], {54}]; t (* Jayanta Basu, Jul 03 2013 *)

a(n) = A075609(n) for n>1. - Alexander Adamchuk, Aug 15 2006

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A075608 a(1) = 1, a(n) = smallest composite number not occurring earlier such that the concatenation a(n-1) and a(n) is a prime.

Original entry on oeis.org

1, 9, 77, 27, 49, 33, 91, 51, 133, 39, 119, 69, 143, 21, 121, 57, 203, 93, 169, 63, 247, 81, 299, 147, 209, 123, 217, 87, 187, 111, 253, 153, 259, 99, 289, 129, 221, 159, 161, 141, 301, 177, 319, 117, 329, 201, 287, 219, 361, 183, 343, 237, 581, 171, 341, 273, 323

Offset: 1

Amarnath Murthy, Sep 28 2002

Cf. A075607, A075609, A075610.

More terms from Michel ten Voorde Jun 23 2003

A075610 a(1) = 1, a(n) = smallest prime number not occurring earlier such that the concatenation a(n-1) and a(n) is a composite number.

Original entry on oeis.org

1, 2, 5, 7, 11, 13, 3, 19, 17, 29, 23, 31, 41, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 89, 103, 101, 109, 107, 113, 127, 131, 137, 139, 149, 151, 167, 157, 173, 163, 179, 181, 191, 193, 197, 199, 223, 227, 211, 233, 229, 239, 241, 251, 269, 263, 271, 257, 277

Offset: 1

Amarnath Murthy, Sep 28 2002

Cf. A075607, A075608, A075609.

More terms from Michel ten Voorde Jun 23 2003

A083439 a(1) = 3; then a(n+1) = smallest prime not already in the sequence such that the concatenations a(n)a(n+1) and a(n+1)a(n) are both primes.

Original entry on oeis.org

3, 7, 19, 13, 61, 151, 31, 139, 67, 37, 79, 193, 163, 127, 157, 97, 43, 103, 307, 367, 457, 643, 73, 277, 223, 229, 373, 199, 109, 313, 211, 271, 241, 421, 181, 283, 397, 337, 349, 331, 577, 523, 463, 613, 439, 541, 571, 433, 787, 547, 661, 769, 487, 601, 823, 709

Offset: 3

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 30 2003

Cf. A073641, A075609.

{ p=3; S=Set(); while(!setsearch(S,p), S=setunion(S,Set([p])); print1(p,", "); forprime(q=2,10^4, if(setsearch(S,q),next); if( isprime(eval(concat(Str(p),Str(q)))) && isprime(eval(concat(Str(q),Str(p)))), p=q; break))) } \\ Max Alekseyev, Apr 24 2009

Corrected and extended by Max Alekseyev, Apr 24 2009

cp's OEIS Frontend

A075607 a(1) = 1, a(n) = smallest number not occurring earlier such that the concatenation a(n-1) and a(n) is a prime.

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A073641 a(1) = 2; a(n) = smallest prime not included earlier such that concatenation of two successive terms is a prime.

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A075608 a(1) = 1, a(n) = smallest composite number not occurring earlier such that the concatenation a(n-1) and a(n) is a prime.

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A075610 a(1) = 1, a(n) = smallest prime number not occurring earlier such that the concatenation a(n-1) and a(n) is a composite number.

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A083439 a(1) = 3; then a(n+1) = smallest prime not already in the sequence such that the concatenations a(n)a(n+1) and a(n+1)a(n) are both primes.

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