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A231207
The prime a(n) = concatenate(A231206(S(n-1)+1 .. S(n))) occurring in the construction of A231206, where S(n) = Sum_{1 <= k <= n} A231206(k).
2
23, 149, 5, 67819, 1011121314151617139, 1820212259, 232425262763, 28293031323371, 3435363738394057, 414243444546474849505152535455565860109, 616264656667686970189, 72737475767778798081179, 8283848586878889909192173
OFFSET
1,1
COMMENTS
The n-th term is the concatenation of the "next" A231206(n) terms of A231206, starting with the (S(n-1)+1)-th term and up to and including the S(n)-th term, where s are the partial sums of A231206, with S(0) = 0, S(1) = A231206(1), etc.
The last term of A231206 used to produce the n-th prime, A231206(S(n)), is usually roughly twice as large as its neighbors, see example. This in turn leads to primes, at the corresponding indices S(n), about twice as long as their neighbors in this sequence.
LINKS
EXAMPLE
Concatenation of the first A(1) = 2 terms of A = A231206, i.e., A(1..2), yields the first prime a(1) = 23. Concatenation of the next A(2) = 3 terms of A231206, i.e., A(3..6), yields a(2) = 149, etc.
The terms A(S(n) := Sum_{1 <= k <= n} A(k)) of A = A231206 in which end the primes a(n), are usually roughly twice as large as their neighbors A(S(n)-1) and A(S(n)+1) which can be chosen as smallest available numbers without the additional constraint of having to yield a prime: A231206 = (..., 4, 9, 5, ... 9, 19, 10, ... 25, 59, 26, ... 27, 63, 28, ... 33, 71, 34, ... 60, 109, 61, ... 70, 198, 72, ... 81, 179, 82, ...), where:
- 9 = A231206(S(2)=5) is twice as large as 4 and 5;
- 5 = A231206(S(3)=6) is exceptionally small: ignore this coincidence;
- 19 = A231206(S(4)=10) is twice as large as 9 and 10,
- 139 = A231206(S(5)=19) is exceptionally large: see below,
- 59 = A231206(S(6)=24) is about twice as large as 25 and 26,
- 63 = A231206(S(7)=30) is about twice as large as 27 and 28,
- 71 = A231206(S(8)=37) is about twice as large as 33 and 34,
- 109 = A231206(S(9)=37) is about twice as large as 60 and 61, etc.
To make the 5th prime, the term A231206(S(5)=19) = 139 had to be chosen much larger than its neighbors (17 and 18), which in turn leads to the 417-digit prime a(19), the concatenation of A231206(173..311). To get this prime, one has to take A231206(311) = 581 again roughly twice as large as its neighbors A231206(310) = 309 and A231206(312) = 310. This leads later again to a prime roughly twice as long as its neighbors in this sequence.
PROG
(PARI) {a=[prefix=2]; remain=pointer=1; for(n=1, 499, my(used=Set(a)); if( !remain, remain=a[pointer++]; print1(prefix", "); prefix=""); for(i=1, 1e9, setsearch(used, i) && next; remain>1 || ispseudoprime( eval( Str( prefix, i ))) || next; prefix=Str(prefix, i); a=concat(a, i); remain--; break ))}
CROSSREFS
Sequence in context: A055827 A255418 A268778 * A142902 A362315 A125333
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Nov 05 2013
STATUS
approved