

A285813


Let p_1<p_2<... be primes such that 2*np_i is prime. a(n) is the smallest i such that 2*n+(odd part of p_i1) 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
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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*n3 is composite then 2*n+p is composite for every prime for which 2*np is prime. Indeed, for these 2*n all such primes p are in the interval (3, 2*n3). Then either 2*np or 2*n+p should be divisible by 3, but 2*np is prime >3. So, 2*n+p is composite.


LINKS

Table of n, a(n) for n=1..86.


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*np), i++; if (isprime(2*n+oddp(p1)), return(i)); ); ); return(0); } \\ Michel Marcus, Apr 29 2017


CROSSREFS

Cf. A284919, A284967, A285770.
Sequence in context: A135062 A088428 A025838 * A236480 A236508 A239000
Adjacent sequences: A285810 A285811 A285812 * A285814 A285815 A285816


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Apr 27 2017


EXTENSIONS

More terms from Michael De Vlieger, Apr 27 2017


STATUS

approved



