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A139324
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Difference between two sequences of primes which indicate two different kinds of places in the prime sequence with some vanishing third order difference.
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1
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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; internal format)
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OFFSET
| 1,1
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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 prime 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).
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EXAMPLE
| 23-19=4=a(1). 41-37=4=a(2). 47-43=4=a(3).
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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
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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}]]
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CROSSREFS
| Sequence in context: A138195 A140744 A179414 * A111655 A175961 A113646
Adjacent sequences: A139321 A139322 A139323 * A139325 A139326 A139327
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KEYWORD
| nonn,less
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 07 2008
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