A199919 - OEIS

%I #27 May 23 2021 06:26:57

%S 3,7,9,15,9,25,9,31,27,37,9,57,9,49,49,63,9,79,9,85,65,49,9,121,27,49,

%T 81,113,9,145,9,127,81,49,69,183,9,49,81,181,9,193,9,169,157,49,9,249,

%U 27,187,81,197,9,241,69,241,81,49,9,337,9,49,209,255,81,289

%N Number of distinct sums of distinct divisors of n when positive and negative divisors are allowed.

%H David A. Corneth, <a href="/A199919/b199919.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernard Jacobson, <a href="http://dx.doi.org/10.1090/S0002-9939-1964-0160746-8">Sums of distinct divisors and sums of distinct units</a>, Proc. Amer. Math. Soc. 15 (1964), 179-183

%H David A. Corneth, <a href="/A199919/a199919.gp.txt">PARI program</a>

%F a(A005153(n)) = 2*sigma(A005153(n)) + 1. - _David A. Corneth_, May 19 2021

%F a(p) = 9 for odd primes p. - _Antti Karttunen_, May 19 2021

%e a(2)=7 because the signed divisors of 2 are -2, -1, 1 and 2 and their all possible sums are -1, -2, -3, 0, 1, 2, 3.

%e a(3)=9 because the signed divisors of 3 are -3, -1, 1 and 3 and their all possible sums are -1, -2, -3, -4, 0, 1, 2, 3, 4.

%t dsdd[n_]:=Module[{divs=Divisors[n]},Length[Union[Total/@Subsets[ Join[ divs,-divs],2Length[divs]]]]]; Array[dsdd,70] (* _Harvey P. Dale_, Jan 19 2015 *)

%o (PARI)

%o A199919(n) = { my(ds=concat(apply(x -> -x,divisors(n)),divisors(n)),m=Map(),s,u=0); for(i=0,(2^#ds)-1,s = sumbybits(ds,i); if(!mapisdefined(m,s), mapput(m,s,s); u++)); (u); }; \\ Slow!

%o sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); }; \\ _Antti Karttunen_, May 19 2021

%o (PARI) See PARI-link \\ _David A. Corneth_, May 20 2021

%Y Cf. A005153, A119347.

%K nonn,look

%O 1,1

%A _Michel Marcus_, Dec 22 2012