A258843 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 + b) = sigma(k).

11, 123, 695, 991, 1919, 2839, 3707, 3841, 7615, 8047, 8055, 9347, 10703, 12847, 16195, 26743, 27089, 32127, 42251, 56419, 59027, 59179, 59389, 59815, 62749, 65113, 74671, 115289, 119211, 122847, 126895, 129495, 168739, 191051, 219295, 224281, 232315, 233729

Offset: 1

Author

Paolo P. Lava, Jun 12 2015

Keywords

• nonn
• base

Comments

It appears that a or b is equal to 1.

The terms that have b=1 are 11, 695, 991, 2839, 3707, 9347, ...; see A232355. - Michel Marcus, Jun 12 2015

If b=1, the number k can be expressed as 2a+b=2a+1. We are looking for numbers that satisfy the relation sigma(a+1)=sigma(k), namely sigma(a+1)=sigma(2a+1). In A232355 we have the numbers such that sigma(k)=sigma((k+1)/2) that match sigma(2a+1)=sigma((2a+1+1)/2)=sigma(a+1). That's why the two "subsequences" are the same thing. - Paolo P. Lava and Michel Marcus, Jun 16 2015

Examples

			11 in base 2 is 1011. If we take 1011 = concat(101,1) then 101 and 1 converted to base 10 are 5 and 1. Finally sigma(5 + 1) = sigma(6) = 12 = sigma(11).
123 in base 2 is 1111011. If we take 1111011 = concat(1,111011) then 1 and 111011 converted to base 10 are 1 and 59. Finally sigma(1 + 59) = sigma(60) = 168 = sigma(123).
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..110

Crossrefs

Cf. A232355, A258813, A258844.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
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+b)=sigma(n) then print(n);
break; fi; fi; od; od; end: P(10^6);

Mathematica

f[n_] := Block[{d = IntegerDigits[n, 2], len, s}, len = Length@ d; s = FromDigits[#, 2] & /@ {Take[d, #], Take[d, -len + #]} & /@ Range[len - 1]; DeleteDuplicates[DivisorSigma[1, #1 + #2] == DivisorSigma[1, n] & @@@ s]]; Select[Range@ 250000, Length@ f@ # > 1 &] (* Michael De Vlieger, Jun 12 2015 *)

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 (sigma(da+db) == sigma(n), return(1));););} \\ Michel Marcus, Jun 12 2015