[seqfan] Re: A093893 Subsequence

[Richard Mathar]

In http://list.seqfan.eu/pipermail/seqfan/2009-May/001521.html Leroy spake

lq> Consider sequence A093893,
lq> This is the list of positive integers n such that the partial sum of any 2 or more divisors of n is composite.
lq> 
lq> What I wonder about is the subsequence, which doesn't seem to be in the EIS, where the nth term is the smallest term of A093893 with exactly n divisors.
lq> 
lq> (Starts at a(2).)
lq> 
lq> 3, 49, 87, etc.
lq> 
lq> It seems that it is very unlikely that this sequence is infinite, or even that it is not short.
lq> 
lq> Can it be proved that this sequence is finite or infinite?

I think this starts 3, 49, 87, 130321, 4753, >1000000, 285541  (n=2 to 8)
The values for n=7 and n=9 to n=31  are all larger than 1 million (if they exist).

isA093893 := proc(n)
	local dvs,s,b2,ps,j ;
	dvs := numtheory[divisors](n) ;
	for s from 2 to 2^nops(dvs)-1 do
		b2 := convert(s,base,2) ;
		if add(b,b=b2) > 1 then
			ps := add( op(j,dvs)*op(j,b2), j=1..nops(b2)) ;
			if isprime(ps) then
				RETURN(false):
			fi;
		fi;
	od:
	RETURN(true) ;
end:

for d from 2 do
	for n from 3 by 2 do
		if numtheory[tau](n) = d then
			if isA093893(n) then
				print(n) ;
				break;
			fi;
		fi;
	od:
od: