A211223 -id:A211223 - cp's OEIS frontend

A246487 Numbers x such that sigma(x) + sigma(R(x)) = sigma(x + R(x)), where R(x) is the digit reversal of x and sigma(x) is the sum of the divisors of x.

78, 87, 104, 401, 1144, 2072, 2178, 2702, 4411, 7038, 7348, 7878, 8307, 8437, 8712, 8787, 11144, 11544, 12584, 15834, 20710, 20913, 21476, 21978, 22164, 26070, 31902, 43851, 44111, 44511, 46122, 48521, 66649, 67412, 87912, 94666, 102786, 122584, 122784, 126984

Offset: 1

Paolo P. Lava, Aug 27 2014

			x = 15834 -> R(x) = 43851 and sigma(15834) + sigma(43851) = 40320 + 59904 = 100224 = sigma(15834 + 43851)= sigma(59685).

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A000203, A004086, A211223.

with(numtheory): P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=n; b:=0;
for k from 1 to ilog10(n)+1 do b:=10*b+(a mod 10); a:=trunc(a/10);
od; if sigma(n)+sigma(b)=sigma(n+b) then print(n); fi;
od; end: P(10^6);

Select[Range[130000],DivisorSigma[1,#]+DivisorSigma[1,IntegerReverse[#]] == DivisorSigma[1,#+IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 27 2017 *)

isok(n) = rn = subst(Polrev(digits(n)), x, 10); sigma(n + rn) == sigma(n) + sigma(rn); \\ Michel Marcus, Aug 29 2014

A218981 Numbers n for which sigma(n) = sigma(w) + sigma(x) + sigma(y) + sigma(z), where n = w + x + y + z, with w, x, y, z all positive.

Original entry on oeis.org

5, 9, 14, 15, 20, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Jon Perry, Nov 08 2012

nonn

Conjecture: This sequence is infinite.

			1 + 1 + 3 + 4 = 9 and sigma(1) + sigma(1) + sigma(3) + sigma(4) = 1 + 1 + 4 + 7 = 13 = sigma(9).

Cf. A000203, A211223, A218852, A218980.

function divisorSum(n) {
c=0;
for (i=1;i<=n;i++) if (n%i==0) c+=i;
return c;
}
ds=new Array();
for (j=1;j<401;j++) ds[j]=divisorSum(j);
a=new Array();
ac=0;
for (j=1;j<100;j++)
for (k=1;k<=j;k++)
for (m=1;m<=k;m++)
for (n=1;n<=m;n++)
if (ds[j]+ds[k]+ds[m]+ds[n]==ds[j+k+m+n]) a[ac++]=j+k+m+n;
a.sort(function(a, b) {return a-b;});
i=0;
while(i++

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

Original entry on oeis.org

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

Felix Huber, Jul 24 2025

nonn

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

			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).

Cf. A001157, A211223, A211225, A381747.

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);

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

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

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A246487 Numbers x such that sigma(x) + sigma(R(x)) = sigma(x + R(x)), where R(x) is the digit reversal of x and sigma(x) is the sum of the divisors of x.

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A218981 Numbers n for which sigma(n) = sigma(w) + sigma(x) + sigma(y) + sigma(z), where n = w + x + y + z, with w, x, y, z all positive.

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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.

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Maple

Mathematica

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