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A139324 Difference between two sequences of primes which indicate two different kinds of places in the prime sequence with some vanishing third order difference. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A138195 A140744 A179414 * A111655 A175961 A113646

Adjacent sequences:  A139321 A139322 A139323 * A139325 A139326 A139327

KEYWORD

nonn,less

AUTHOR

Roger L. Bagula, Jun 07 2008

STATUS

approved

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)