[seqfan] Re: A093893 Subsequence

[Maximilian Hasler]

If we require 7 divisors, then the number must be of the form n=p^6
and the divisors are p^k, k=0..6
and the partial sums are always composite if they don't start with 1
or if they have an even number of terms
so we need only to check if any sum of 3 or 5 of the divisors,
necessarily including 1, is composite.

? a(3)
%52 = 49

? a(5)
%53 = 130321

? a(7)
%54 = 7212549413161

? a(11)
%55 = 174913992535407978606601

Maximilian

a(nd)={ isprime(nd)|error("version limited to prime args");
forprime(p=3,default(primelimit),forstep(i=2,nd-1,2, /*choose i among
the nd-1 divisors p^1,...,p^(nd-1) and check if 1 + their sum is prime
*/ forvec( d=vector(i,j,[1,nd-1]),
isprime(sum(j=1,i,p^d[j],1))&next(3),2));return(p^(nd-1)))}



On Mon, May 25, 2009 at 12:21 PM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:
>
> 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:
>
>
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