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A386205 Numbers k for which a solution to sigma_2(x) + sigma_2(k-x) = sigma_2(k) in positive integers exists.

%I A386205 #16 Aug 04 2025 22:45:56
%S A386205 100,155,434,465,639,700,783,866,875,1085,1100,1300,1395,1700,1705,
%T A386205 1900,2015,2170,2300,2625,2900,3100,3255,3565,3700,4100,4123,4185,
%U A386205 4300,4473,4495,4700,4774,4900,5115,5300,5642,5735,5900,6045,6062,6100,6355,6665,6700,7100
%N A386205 Numbers k for which a solution to sigma_2(x) + sigma_2(k-x) = sigma_2(k) in positive integers exists.
%C A386205 Since sigma_2(n) is multiplicative, for every prime p>5, 100*p is a term. In other words, for every prime p>5, sigma_2(100*p) = sigma_2(4)*sigma_2(p) + sigma_2(96)*sigma_2(p). - _Ivan N. Ianakiev_, Jul 29 2025
%e A386205 100 is a term because sigma_2(4) + sigma_2(96) = 21 + 13650 = 13671 = sigma_2(100).
%e A386205 465 is a term because sigma_2(57) + sigma_2(408) = 3620 + 246500 = 250120 = sigma_2(465).
%p A386205 with(NumberTheory):
%p A386205 A:=proc(n)
%p A386205     option remember;
%p A386205     local k,x;
%p A386205     if n=1 then
%p A386205         100
%p A386205     else
%p A386205         for k from procname(n-1)+1 do
%p A386205             for x to k/2 do
%p A386205                 if sigma[2](x)+sigma[2](k-x)=sigma[2](k) then
%p A386205                     return k
%p A386205                 fi
%p A386205             od
%p A386205         od
%p A386205     fi;
%p A386205 end proc;
%p A386205 seq(A(n),n=1..5);
%t A386205 f[n_]:=Select[Range[n/2],DivisorSigma[2,#]==DivisorSigma[2,n]-DivisorSigma[2,n-#]&]; Select[Range[4100],f[#]!={}&] (* _Ivan N. Ianakiev_, Jul 29 2025 *)
%o A386205 (PARI) isok(k) = my(sk2=sigma(k,2)); for (i=1, k-1, if (sigma(i,2) + sigma(k-i,2) == sk2, return(1))); \\ _Michel Marcus_, Jul 29 2025
%Y A386205 Cf. A001157, A211223, A211225, A381747.
%K A386205 nonn
%O A386205 1,1
%A A386205 _Felix Huber_, Jul 24 2025