[seqfan] Re: Question about abundant numbers (A5101)

[Richard J. Mathar]

aw> Date: Mon, 12 Dec 2022 21:35:38 -0500
aw> From: Allan Wechsler <acwacw at gmail.com>
aw> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
aw> Subject: [seqfan] Re: Question about abundant numbers (A5101)
aw> 
aw> This line of thinking leads me to wonder whether anyone has investigated
aw> the number of distinct values that partial sums of the divisors of N can
aw> take on.
aw> 
aw> This starts out looking a lot like https://oeis.org/A107748, but then
aw> A107748(10) = 14, where the sequence I'm thinking of ought to have 16.

This is A308605. The zero (sum of the empty subset of the set  of divisors)
counts as a distinct value. In Maple:

A308605 := proc(n)
    # set of the divisors
    dvs := numtheory[divisors](n) ;
    # set of all the subsets fo the divisors
    pdivs := combinat[powerset](dvs) ;
    # the set of the sums in subsets of divisors
    dvss := {} ;
    # loop over all subsets of divisors
    for s in pdivs do
        # compute sum over entries of the subset
        sps := add(d,d=s) ;
        # add sum to the realized set of sums
        dvss := dvss union {sps} ;
    end do:
    # count number of distinct entries (distinct sums)
    nops(dvss) ;
end proc:

seq(A308605(n),n=1..10) ;

Obviously one can also get this via A119347.