A128655 -id:A128655 - cp's OEIS frontend

A126938 a(1) = 3, a(n) = the smallest prime p > a(n-1) such that (a(n-1)+p)/2 is prime.

3, 7, 19, 43, 79, 127, 151, 163, 199, 223, 331, 367, 379, 439, 487, 607, 619, 643, 739, 883, 991, 1051, 1087, 1171, 1231, 1327, 1471, 1627, 1699, 1747, 1759, 1987, 1999, 2179, 2383, 2551, 2683, 2731, 2767, 3067, 3259, 3343, 3571, 3643, 3739, 3847, 3907

Offset: 1

Zak Seidov, Mar 18 2007

nonn

Starting with a(2)=7 all terms are 7 mod 12. - Zak Seidov, Feb 26 2017

			(3+7)/2=5 prime, (7+19)/2=13 prime, (19+43)/2=31 prime, etc.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A084704, A128653, A128654, A128655.

A[1]:= 3: A[2]:= 7:
for n from 3 to 100 do A[n]:= f(A[n-1]) od:
seq(A[i],i=1..100); # Robert Israel, Feb 27 2017

s={3};pn=3;n=PrimePi[pn];Do[Do[p=Prime[i];If[PrimeQ[(pn+p)/2],AppendTo[s,p];pn=p;n=i;Break[]],{i,n+1,10000}],{112}];s
sp[n_]:=Module[{p=NextPrime[n]},While[!PrimeQ[(n+p)/2],p=NextPrime[p]];p]; NestList[sp,3,50] (* Harvey P. Dale, Apr 12 2013 *)

step(q)=forprime(p=q+1,, if(isprime((p+q)/2), return(p)))
first(n)=my(v=vector(n)); v[1]=3; for(k=2,n, v[k]=step(v[k-1])); v \\ Charles R Greathouse IV, Feb 27 2017

A128653 Primes occurring in A084704 exactly twice.

Original entry on oeis.org

43, 53, 59, 79, 113, 151, 163, 167, 197, 229, 269, 313, 331, 359, 367, 397, 419, 421, 449, 541, 607, 617, 619, 683, 719, 739, 757, 857, 883, 887, 907, 911, 997, 1021, 1051, 1063, 1103, 1117, 1193, 1249, 1289, 1303, 1427, 1433, 1459, 1471, 1489, 1493, 1499

Offset: 1

Zak Seidov, Mar 18 2007

nonn

			43 occurs as the 7th and 10th terms in A084704: (prime(8)+43)/2 = (19+43)/2 = 31 prime and (prime(11)+43)/2 = (31+43)/2 = 37 prime;
53 is 9th and 12th terms in A084704: (prime(10)+53)/2 = (29+53)/2 = 41 prime and (prime(13)+53)/2 = (41+53)/2 = 47 prime.

For primes occurring in A084704 exactly 3 and 4 times see A128654, A128655. Cf. A084704 Smallest prime p > prime(n) such that (p+prime(n))/2 is prime.

A128654 Primes occurring in A084704 exactly thrice.

Original entry on oeis.org

263, 821, 953, 1049, 1597, 2143, 2273, 2677, 2683, 2699, 2749, 3251, 3389, 3709, 3739, 3761, 3929, 4013, 4243, 4409, 4519, 4603, 4621, 4639, 4657, 4943, 5009, 5387, 5449, 5483, 5783, 6173, 6311, 6373, 6703, 6737, 6869, 6983, 7517, 7603, 7717, 7993

Offset: 1

Zak Seidov, Mar 18 2007

nonn

			263 occurs as the 42nd, 51st and 53rd terms: (prime(#)+263)/2/(AT){43,52,54}= {227,251,257} primes.

Cf. A084704, A128653, A128655.

cp's OEIS Frontend

A126938 a(1) = 3, a(n) = the smallest prime p > a(n-1) such that (a(n-1)+p)/2 is prime.

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A128653 Primes occurring in A084704 exactly twice.

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A128654 Primes occurring in A084704 exactly thrice.

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