A139324 Difference between two sequences of primes which indicate two different kinds of places in the prime sequence with some vanishing third-order difference.

4, 4, 4, 4, 6, 4, 6, 4, 4, 6, 6, 6, 8, 6, 4, 4, 6, 8, 8, 6, 6, 4, 4, 4, 4, 6, 4, 6, 4, 6, 4, 8, 6, 4, 4, 6, 4, 10, 4, 6, 4, 6, 18, 12, 4, 4, 6, 6, 4, 6, 6, 8, 10, 12, 8, 6, 4, 6, 6, 8, 4, 12, 4, 4, 6, 6, 8, 4, 4, 4, 4, 6, 12

Offset

1,1

Comments

There are two sequences of primes at which two third-order differences vanish:

one is b(n) = 23, 41, 47, 71, 89, 233, ... which contains all primes prime(n) such that prime(n-2) - 3*prime(n-1) + 3*prime(n) - prime(n+1) = 0;

the other is A139313(n) = 19, 37, 43, ... such that -prime(n-1) + 3*prime(n) - 3*prime(n+1) - prime(n+2) = 0.

Then by definition a(n) = b(n) - A139313(n).

Links

Table of n, a(n) for n=1..73.

Example

23 - 19 = 4 = a(1). 41 - 37 = 4 = a(2). 47 - 43 = 4 = a(3).

Maple

A139324a := proc(n) if n = 1 then 23; else a := nextprime(procname(n-1)) ; while (true ) do if prevprime(prevprime(a))-3*prevprime(a)+3*a-nextprime(a) =0 then return a; end if; a := nextprime(a) ; end do: end if; end proc:
A139313 := proc(n) if n = 1 then 19; else a := nextprime(procname(n-1)) ; while (true ) do if -prevprime(a)+3*a-3*nextprime(a)+nextprime(nextprime(a)) = 0 then return a; end if; a := nextprime(a) ; end do: end if; end proc:
A139324 := proc(n) A139324a(n)-A139313(n) ; end proc:
seq(A139324(n), n=1..80) ; # R. J. Mathar, Jun 15 2011

Mathematica

Flatten[Table[If[ Prime[ -2 +n] - 3 Prime[ -1 + n] + 3 Prime[n] - 1 Prime[1 + n] == 0, Prime[n], {}], {n, 3, 500}]] - Flatten[ Table[If[ -Prime[ -1 + n] + 3*Prime[n] - 3*Prime[1 + n] + Prime[n + 2] == 0, Prime[n], {}], {n, 2, 500}]]

Crossrefs

Sequence in context: A140744 A179414 A361248 * A111655 A175961 A113646

Adjacent sequences: A139321 A139322 A139323 * A139325 A139326 A139327

Keyword

nonn,less

Author

Roger L. Bagula, Jun 07 2008

Status

approved