A308605 - OEIS

%I #40 Nov 29 2024 15:07:02

%S 2,4,4,8,4,13,4,16,8,16,4,29,4,16,16,32,4,40,4,43,16,16,4,61,8,16,16,

%T 57,4,73,4,64,16,16,16,92,4,16,16,91,4,97,4,64,56,16,4,125,8,64,16,64,

%U 4,121,16,121,16,16,4,169,4,16,60,128,16,145,4,64,16,143,4,196,4,16,64,64,16,169,4,187,32,16,4,225

%N Number of distinct subset sums of the subsets of the set of divisors of n. Here 0 (as the sum of an empty subset) is included in the count.

%C Conjecture: When the terms are sorted and the duplicates deleted a supersequence of A030058 is obtained. Note that A030058 is a result of the same operation applied to A030057.

%C a(p^k) = 2^(k+1) if p is prime, and a(n) = 2n+1 if n is an even perfect number. - _David Radcliffe_, Dec 22 2022

%H Antti Karttunen, <a href="/A308605/b308605.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 + A119347(n). - _Rémy Sigrist_, Jun 10 2019

%e The subsets of the set of divisors of 6 are {{},{1},{2},{3},{6},{1,2},{1,3},{1,6},{2,3},{2,6},{3,6},{1,2,3},{1,2,6},{1,3,6},{2,3,6},{1,2,3,6}}, with the following sums {0,1,2,3,6,3,4,7,5,8,9,6,9,10,11,12}, of which 13 are distinct. Therefore, a(6)=13.

%p A308605 := proc(n)

%p     # set of the divisors

%p     dvs := numtheory[divisors](n) ;

%p     # set of all the subsets of the divisors

%p     pdivs := combinat[powerset](dvs) ;

%p     # the set of the sums in subsets of divisors

%p     dvss := {} ;

%p     # loop over all subsets of divisors

%p     for s in pdivs do

%p         # compute sum over entries of the subset

%p         sps := add(d,d=s) ;

%p         # add sum to the realized set of sums

%p         dvss := dvss union {sps} ;

%p     end do:

%p     # count number of distinct entries (distinct sums)

%p     nops(dvss) ;

%p end proc:

%p seq(A308605(n),n=1..20) ; # _R. J. Mathar_, Dec 20 2022

%t f[n_]:=Length[Union[Total/@Subsets[Divisors[n]]]]; f/@Range[100]

%o (Python)

%o from sympy import divisors

%o def a308605(n):

%o     s = set([0])

%o     for d in divisors(n):

%o         s = s.union(set(x + d for x in s))

%o     return len(s) # _David Radcliffe_, Dec 22 2022

%o (PARI) A308605(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c); }; \\ (after _Chai Wah Wu_'s Python-code in A119347, but see also above) - _Antti Karttunen_, Nov 29 2024

%o (PARI) A308605(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=0,poldegree(p),(0<polcoeff(p, i))); }; \\ _Antti Karttunen_, Nov 29 2024

%Y One more than A119347.

%Y Cf. A030057, A030058, A083207 (positions of odd terms), A179527 (parity of terms).

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Jun 10 2019

%E Definition clarified and more terms added by _Antti Karttunen_, Nov 29 2024