A258813 Numbers k with the property that it is possible to write the base 2 expansion of k as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have sigma(a) + sigma (b) = sigma(k) - k.

9, 15, 27, 39, 51, 77, 143, 207, 329, 377, 473, 611, 903, 1241, 1243, 1273, 1437, 1591, 2117, 2303, 2975, 4189, 8401, 8657, 11993, 13849, 15611, 16771, 18239, 18599, 19359, 25331, 28877, 37291, 41747, 41807, 61549, 67037, 72601, 82169, 83411, 83711, 87449, 99329

Offset: 1

Author

Paolo P. Lava, Jun 11 2015

Keywords

• nonn
• base

Examples

			9 in base 2 is 1001. If we take 1001 = concat(10,01) then 10 and 01 converted to base 10 are 2 and 1. Finally sigma(2) + sigma(1) = sigma(9) - 9 = 4.
180953 in base 2 is 101100001011011001. If we take 101100001011011001 = concat(101100001011,011001) then 101100001011 and 011001 converted to base 10 are 2827 and 25. Finally sigma(2827) + sigma(25) = sigma(180953) - 180953 = 3127.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..80

Crossrefs

Cf. A253824, A253825, A258843, A258844.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,j,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
j:=0; for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
if a*b>0 then if sigma(a)+sigma(b)=sigma(n)-n then print(n);
break; fi; fi; od; od; end: P(10^9);

PARI

isok(n) = {b = binary(n); if (#b > 1, for (k=1, #b-1, vba = Vecrev(vector(k, i, b[i])); vbb = Vecrev(vector(#b-k, i, b[k+i])); da = sum(i=1, #vba, vba[i]*2^(i-1)); db = sum(i=1, #vbb, vbb[i]*2^(i-1)); if (da && db && (sigma(da)+sigma(db) == sigma(n)-n), return(1));););} \\ Michel Marcus, Jun 13 2015