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A285813
Let p_1<p_2<... be primes such that 2*n-p_i is prime. a(n) is the smallest i such that 2*n+(odd part of p_i-1) is prime or a(n)=0 if there is no such i.
0
0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 4, 3, 6, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 10, 1, 1, 1, 1, 4, 5, 1, 1, 1, 2, 2, 1, 5, 3, 6, 3, 1, 1, 1, 2, 1, 1, 2, 1, 9, 6, 0, 6, 2, 5, 2, 1, 1, 4, 2, 1, 7, 4, 4, 7, 1, 2, 8, 3, 7, 1, 2, 4, 1, 1, 1, 2, 2, 1
OFFSET
1,7
COMMENTS
The sequence of positions of zeros either grows very fast or is finite. We are inclined to the latter option (cf. our comments in A284919 and in A285770). By A284967, the first three positions of zeros are {4,59,434} and, according to the last calculations by Michel Marcus, no more positions up to 5*10^7.
There are many more terms in A284919 than zeros in this sequence. The reason of this phenomenon is the following. In A284919, if n is not divisible by 3 and 2*n-3 is composite then 2*n+p is composite for every prime for which 2*n-p is prime. Indeed, for these 2*n all such primes p are in the interval (3, 2*n-3). Then either 2*n-p or 2*n+p should be divisible by 3, but 2*n-p is prime >3. So, 2*n+p is composite.
MATHEMATICA
Flatten@ Table[FirstPosition[#, p_ /; PrimeQ@ p] /. k_ /; MissingQ@ k -> {0} &@ Map[2 n + NestWhile[#/2 &, # - 1, EvenQ@ # &] &, Select[Prime@ Range@ PrimePi[2 n - 2], PrimeQ[2 n - #] &]], {n, 86}] (* Michael De Vlieger, Apr 27 2017, Version 10.2 *)
PROG
(PARI) oddp(n) = n/2^valuation(n, 2);
a(n) = {i = 0; forprime(p=2, 2*n, if (isprime(2*n-p), i++; if (isprime(2*n+oddp(p-1)), return(i)); ); ); return(0); } \\ Michel Marcus, Apr 29 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Apr 27 2017
EXTENSIONS
More terms from Michael De Vlieger, Apr 27 2017
STATUS
approved