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

100, 155, 434, 465, 639, 700, 783, 866, 875, 1085, 1100, 1300, 1395, 1700, 1705, 1900, 2015, 2170, 2300, 2625, 2900, 3100, 3255, 3565, 3700, 4100, 4123, 4185, 4300, 4473, 4495, 4700, 4774, 4900, 5115, 5300, 5642, 5735, 5900, 6045, 6062, 6100, 6355, 6665, 6700, 7100

Offset: 1

Author

Felix Huber, Jul 24 2025

Keywords

• nonn

Comments

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

Examples

			100 is a term because sigma_2(4) + sigma_2(96) = 21 + 13650 = 13671 = sigma_2(100).
465 is a term because sigma_2(57) + sigma_2(408) = 3620 + 246500 = 250120 = sigma_2(465).
		

Crossrefs

Cf. A001157, A211223, A211225, A381747.

Programs

Maple

with(NumberTheory):
A:=proc(n)
    option remember;
    local k,x;
    if n=1 then
        100
    else
        for k from procname(n-1)+1 do
            for x to k/2 do
                if sigma[2](x)+sigma[2](k-x)=sigma[2](k) then
                    return k
                fi
            od
        od
    fi;
end proc;
seq(A(n),n=1..5);

Mathematica

f[n_]:=Select[Range[n/2],DivisorSigma[2,#]==DivisorSigma[2,n]-DivisorSigma[2,n-#]&]; Select[Range[4100],f[#]!={}&] (* Ivan N. Ianakiev, Jul 29 2025 *)

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