A240563 Primes formed from concatenation of higher primes onto the previous entry until prime, starting from 2.

2, 23, 2311, 231131, 23113147, 23113147229, 23113147229251, 23113147229251577, 23113147229251577857, 23113147229251577857859, 23113147229251577857859911, 231131472292515778578599111123, 2311314722925157785785991111231223

Offset: 1

Author

Bill McEachen, Apr 07 2014

Keywords

• nonn
• base

Comments

This generates a monotonically increasing sequence, nicely spread out, likely infinite. By altering the starting prime value, a family of such sequences can easily be generated.

Derived from A080155. - T. D. Noe, Apr 11 2014

From the first 155 points, with x = #digits, y = sequence pointer y~ A*x^B with (A, B) = (0.6624, 0.8106). This indicates a 100-digit prime in the vicinity of y = 28 for example. - Bill McEachen, Apr 13 2014

Only from the first 100 entries, it would appear that an upper bound on the number of digits in a(n) is A092777(n). - Bill McEachen, Sep 15 2015

Examples

			Begin from 2.
Next we try 23 - it is prime, this sets next iteration (23 is the "constant" part), upon which we try higher primes.
Next we try 235 - composite; next we try 237 - composite; next we try 2311 - prime, this sets next iteration (2311 now becomes the "constant" part), upon which we try higher primes.
Next we try 231113 - composite; next we try 231117 - composite; ...; next we try 231131 - prime, this sets next iteration (231131 now becomes the "constant" part), upon which we try higher primes.
Next we try 23113147 - prime, this sets next iteration (23113147 now becomes the "constant" part), upon which we try higher primes.
		

Links

• Bill McEachen, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A069151 (variant).

Cf. A080155 (primes used in concatenation).

Programs

Maple

X:= 2: p:= 3: a[1]:= 2:
for i from 2 to 30 do
  while not isprime(X*10^(1+ilog10(p))+p) do
     p:= nextprime(p)
  od:
  X:= X*10^(1+ilog10(p))+p;
  a[i]:= X;
  p:= nextprime(p);
od:
seq(a[i],i=1..30); # Robert Israel, Sep 15 2015

Mathematica

s[1] = 2; s[n_] := s[n] = Block[{d = Flatten[IntegerDigits /@ Array[s, n-1]], p = NextPrime@s[n - 1]}, While[! PrimeQ@ FromDigits@ Join[d, IntegerDigits@p], p = NextPrime@p]; p]; a[n_] := FromDigits@ Flatten[ IntegerDigits /@ Array[s, n]]; Array[a, 10] (* Giovanni Resta, Apr 09 2014 *)

PARI

print1(N=2); p=3; for(n=2,10, while(!isprime(eval(Str(N,p))), p=nextprime(p+1)); N=eval(Str(N,p)); p=nextprime(p+1); print1(", "N)) \\ Charles R Greathouse IV, Apr 09 2014

Extensions

a(7)-a(13) from Giovanni Resta, Apr 09 2014