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Even semiprimes that are the exact average of four consecutive odd semiprimes.
2

%I #24 Sep 25 2023 07:29:05

%S 74,146,194,218,302,482,586,734,746,842,914,1042,1138,1262,1346,1438,

%T 1574,1646,1654,1838,1874,1894,1906,1942,2026,2186,2206,2458,2462,

%U 2762,2906,2962,2974,3098,3106,3202,3218,3826,4198,4274,4286,4322,4414,4502,4534,4622,4666,4754,4934,4946

%N Even semiprimes that are the exact average of four consecutive odd semiprimes.

%e 74 is a term because (65 + 69 + 77 + 85)/4 = 74 = 2*37 is an even semiprime.

%e 218 is a term because (215 + 217 + 219 + 221)/4 = 218 = 2*109 is an even semiprime.

%t sp=Select[Range[5,5200,2], PrimeOmega[#]==2&]; a={}; For[i=1, i<Length[sp]-3, i++, hav=Sum[Part[sp,k],{k,i,i+3}]/8; If[PrimeQ[hav], AppendTo[a,2hav]]]; a (* _Stefano Spezia_, Aug 25 2023 *)

%o (PARI) upto(n) = {my(res = List(), l = List([0, 9, 15, 21]), s = sum(i = 1, #l, l[i]), i = l[#l]+2, ntimes4 = 4*n); while(1, if(bigomega(i) == 2, s += (i-l[1]); if(s > ntimes4, return(res)); if(s % 8 == 0 && isprime(s/8), listput(res, s/4)); listpop(l, 1); listput(l, i)); i+=2)} \\ _David A. Corneth_, Aug 26 2023

%Y Cf. A046315, A365200, A365202.

%Y Subset of A100484.

%K nonn

%O 1,1

%A _Elmo R. Oliveira_, Aug 25 2023