cp's OEIS Frontend

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.

%I A258813 #21 Jul 07 2025 18:27:18
%S A258813 9,15,27,39,51,77,143,207,329,377,473,611,903,1241,1243,1273,1437,
%T A258813 1591,2117,2303,2975,4189,8401,8657,11993,13849,15611,16771,18239,
%U A258813 18599,19359,25331,28877,37291,41747,41807,61549,67037,72601,82169,83411,83711,87449,99329
%N 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.
%H A258813 Paolo P. Lava, <a href="/A258813/b258813.txt">Table of n, a(n) for n = 1..80</a>
%e A258813 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.
%e A258813 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.
%p A258813 with(numtheory): P:=proc(q) local a,b,c,j,k,n;
%p A258813 for n from 1 to q do c:=convert(n,binary,decimal);
%p A258813 j:=0; for k from 1 to ilog10(c) do
%p A258813 a:=convert(trunc(c/10^k),decimal,binary);
%p A258813 b:=convert((c mod 10^k),decimal,binary);
%p A258813 if a*b>0 then if sigma(a)+sigma(b)=sigma(n)-n then print(n);
%p A258813 break; fi; fi; od; od; end: P(10^9);
%o A258813 (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
%Y A258813 Cf. A253824, A253825, A258843, A258844.
%K A258813 nonn,base
%O A258813 1,1
%A A258813 _Paolo P. Lava_, Jun 11 2015