Consider the series of semiprimes. Now consider the series of numbers representing the Sum Of Digits (SOD) of each semiprime.

(1.) How many times does the number '9' occur, and why?

(2.) What is the smallest semiprime that has the same SOD as the next consecutive semiprime? or prove that there is no pair of consecutive semiprimes that have the same SOD.

I used the following Mathematica code to find 4 such semiprimes

SemiQ[n_]:=(

divs=Divisors[n];

If[Length[divs]ƒú3 || (Length[divs]ƒú4 && Mod[divs[[3]],divs[[2]]]„j0),

Return[True],

Return[False]];);

Sod[n_]:=Total[IntegerDigits[n]];

n=2;

prevSod=0;

prevSp=0;

While[n„T1000,

If[SemiQ[n],

s=Sod[n];

If[sƒúprevSod,

Print[prevSp," ",n];

];

prevSod=s;

prevSp=n;

];

n++;

];

and they are:
146, 155
226, 235
278, 287
346, 355

Posted by Daniel on 2009-09-08 12:57:22