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A379162 Ulam numbers that are sphenics.

%I A379162 #14 Jan 03 2025 21:23:24
%S A379162 102,114,138,182,238,258,273,282,370,402,429,434,483,602,627,646,861,
%T A379162 986,1023,1030,1311,1335,1338,1406,1462,1790,1834,1902,1946,2054,2093,
%U A379162 2134,2247,2330,2354,2445,2486,2613,2630,2635,2674,2919,2985,3070,3219,3395
%N A379162 Ulam numbers that are sphenics.
%C A379162 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.
%H A379162 Robert Israel, <a href="/A379162/b379162.txt">Table of n, a(n) for n = 1..10000</a>
%e A379162 102 is a term because 102=2*3*17 is the product of 3 distinct primes and 102 is an Ulam number.
%e A379162 114 is a term because 114=2*3*19 is the product of 3 distinct primes and 114 is an Ulam number.
%e A379162 273 is a term because 273=3*7*13 is the product of 3 distinct primes and 273 is an Ulam number.
%p A379162 N:= 10^4: # for terms <= N
%p A379162 U:= [1, 2]: V:= Vector(N): V[3]:= 1: R:= NULL:
%p A379162 for i from 3 do
%p A379162    for k from U[-1]+1 to N do
%p A379162      if V[k] = 1 then
%p A379162        J:= select(`<=`, U +~ k, N);
%p A379162        V[J]:= V[J] +~ 1;
%p A379162        U:= [op(U), k];
%p A379162        F:= ifactors(k)[2]:
%p A379162        if F[.., 2] = [1, 1, 1] then R:= R, k; break fi
%p A379162    od;
%p A379162    if k > N then break fi;
%p A379162 od:
%p A379162 R; # _Robert Israel_, Jan 03 2025
%t A379162 seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, FactorInteger[#][[;; , 2]] == {1, 1, 1} &]]; seq[300] (* _Amiram Eldar_, Dec 17 2024, after _Jean-François Alcover_ at A002858 *)
%Y A379162 Intersection of A002858 and A007304.
%Y A379162 Cf. A068820, A378795.
%K A379162 nonn
%O A379162 1,1
%A A379162 _Massimo Kofler_, Dec 17 2024