A213025 Balanced semiprimes (of order one): semiprimes which are the average of the previous semiprime and the following semiprime.

34, 86, 94, 122, 142, 185, 194, 202, 214, 218, 262, 289, 302, 314, 321, 358, 371, 394, 407, 413, 415, 422, 446, 471, 489, 493, 497, 517, 535, 562, 581, 586, 626, 634, 669, 687, 698, 734, 785, 791, 815, 838, 842, 922, 982, 989, 1042, 1057, 1079, 1135, 1138

Offset: 1

Author

Gerasimov Sergey, Jun 03 2012

Keywords

• nonn

Comments

Semiprimes that are the average of three successive semiprimes.

First term not also in A086005 is 185. - Alonso del Arte, Jun 04 2012

Examples

			194 is in the sequence because 194 = (187 + 194 + 201)/3 = (A001358(61) + A001358(62) + A001358(63))/3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A086005 (subsequence), A001358, A006562, A065516, A212820.

Programs

Haskell

a213025 n = a213025_list !! (n-1)
a213025_list = f a001358_list where
   f (x:sps'@(y:z:sps)) | 2 * y == (x + z) = y : f sps'
                        | otherwise        = f sps'
-- Reinhard Zumkeller, Jun 10 2012

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 s;
      s:= `if`(n=1, 4, a(n-1));
      do s:= nextsp(s);
         if s=(prevsp(s)+nextsp(s))/2 then break fi
      od; s
    end:
seq (a(n), n=1..100);  # Alois P. Heinz, Jun 03 2012

Mathematica

bspQ[{a_,b_,c_}]:=b==(a+c)/2; With[{sp=Select[Range[1200],PrimeOmega[#] == 2&]}, Transpose[Select[Partition[sp,3,1],bspQ]][[2]]] (* Harvey P. Dale, Nov 18 2012 *)
Select[Partition[Select[Range[1200],PrimeOmega[#]==2&],3,1],Mean[#]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Jul 31 2025 *)

Formula

2*sp_(n) = sp_(n - 1) + sp_(n + 1).

a(n) = (1/3) * (sp(i) + sp(i + 1) + sp(i + 2)), for some i(n).