A212820 Balanced primes which are the average of two successive semiprimes.

5, 53, 173, 211, 1511, 3307, 3637, 4457, 4993, 6863, 11411, 11731, 11903, 12653, 15907, 18223, 20107, 20201, 20347, 20731, 22051, 23801, 26041, 35911, 39113, 40493, 46889, 47303, 51551, 52529, 60083, 63559, 69623, 71011, 75787, 77081, 78803, 85049, 91297

Offset: 1

Author

Gerasimov Sergey, May 28 2012

Keywords

• nonn

Comments

Prime p which is the average of the previous prime and the following prime and is also the average of two successive semiprimes.

Examples

			53 is in the sequence because it is the average of 47 and 59 (the two neighboring primes) and 51 and 55 (the two neighboring semiprimes).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006562, A103654.

Programs

Maple

with(numtheory):
prevsp:= proc(n) local k; for k from n-1 by -1
           while isprime(k) or bigomega(k)<>2 do od; k end:
nextsp:= proc(n) local k; for k from n+1
           while isprime(k) or bigomega(k)<>2 do od; k end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 2, a(n-1));
      do p:= nextprime(p);
         if p=(prevprime(p)+nextprime(p))/2 and
            p=(prevsp(p)+nextsp(p))/2 then break fi
      od; p
    end:
seq (a(n), n=1..40);  # Alois P. Heinz, Jun 03 2012

Mathematica

prevsp[n_] := Module[{k}, For[k = n-1, PrimeQ[k] || PrimeOmega[k] != 2, k--]; k];
nextsp[n_] := Module[{k}, For[k = n+1, PrimeQ[k] || PrimeOmega[k] != 2 , k++]; k];
a[n_] := a[n] = Module[{p}, p = If[n==1, 2, a[n-1]]; While[True, p = NextPrime[p]; If[p == (NextPrime[p, -1] + NextPrime[p])/2 && p == (prevsp[p] + nextsp[p])/2, Break[]]]; p];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

Formula

{ A212820 } = { A006562 } intersection { A103654 }.