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Irregular array read by rows, where the n-th row lists the primes p < A002110(n) such that 2*A002110(n) - p is also prime.
2

%I #89 Nov 04 2015 12:50:00

%S 5,7,13,17,19,23,29,11,19,23,31,37,41,47,53,61,67,71,73,83,89,103,107,

%T 109,113,127,137,139,149,151,157,163,179,181,191,193,197,17,23,29,37,

%U 53,59,71,73,97,101,103,107,113,127,137,139,157,163,173,179,197,199,211,223,229,257,263,271

%N Irregular array read by rows, where the n-th row lists the primes p < A002110(n) such that 2*A002110(n) - p is also prime.

%C For each p in row n, (p, 2*A002110(n) - p) is a pair of centered primes at A002110(n) = prime(n)#.

%C The number of terms in row n are 0, 1, 6, 30, 190, ...; this is A147517.

%H Jean-Marc Rebert, <a href="/A260405/b260405.txt">Table of n, a(n) for n = 1..18866</a> (the first 7 rows).

%e The first row is empty.

%e T(2,1) = 5 because (5,7) is a pair of primes centered at A002110(2) = prime(2)# = 6.

%e Triangle starts:

%e [];

%e [5];

%e [7, 13, 17, 19, 23, 29];

%e [11, 19, 23, 31, 37, 41, 47, 53, 61, 67, 71, 73, 83, 89, 103, 107, 109, 113, 127, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197];

%e ...

%o (PARI) a147517=[0,1,6,30,190,1564,17075,226758,3792532]

%o a002110(n)=prod(i=1, n, prime(i))

%o a(n)=my(k=2,maxk=2,primorielle=2,s=0,y=5);if(n==1, y=5, maxk=2;while(sum(k=2, k=maxk, a147517[k])<n, maxk++; s=sum(k=2, k=maxk, a147517[k-1]));k=maxk; primorielle=a002110(k); forprime(p=2, primorielle, if(isprime(2*primorielle-p), s++; if(s==n, y=p;break)));return(y);)

%o (PARI) row(n) = {v = []; pn = prod(i=1, n, prime(i)); forprime(p=1, pn-1, if (isprime(2*pn-p), v = concat(v, p))); v;} \\ _Michel Marcus_, Aug 02 2015

%o (PARI) a002110(n) = prod(p=1,n,prime(i));

%o T(n,k) = my(P= a002110(n),compteur = 0,q=0,y=-1);forprime(p=1,P-1,q = 2*P-p;if(isprime(q),compteur++;if(compteur== k,y=p;y;break)));y

%Y Cf. A000040, A002110, A147517, A260384.

%Y Cf. A260388 (row 3).

%K nonn,tabf

%O 1,1

%A _Jean-Marc Rebert_, Jul 24 2015