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

1,1

Comments

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

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A006562, A103654.

Sequence in context: A094847 A001992 A139899 * A094849 A094852 A267543

Adjacent sequences: A212817 A212818 A212819 * A212821 A212822 A212823

Keyword

nonn

Author

Gerasimov Sergey, May 28 2012

Status

approved