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A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

%I A237290 #37 Nov 29 2024 14:45:59
%S A237290 1,6,8,28,12,78,16,120,52,144,24,406,28,192,192,496,36,780,40,903,256,
%T A237290 288,48,1830,124,336,320,1596,60,2628,64,2016,384,432,384,4186,76,480,
%U A237290 448,4095,84,4656,88,2688,2184,576,96,7750,228,2976,576,3136,108,7260
%N A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.
%H A237290 Antti Karttunen, <a href="/A237290/b237290.txt">Table of n, a(n) for n = 1..20000</a> (first 200 terms from Vincenzo Librandi)
%H A237290 Jon Maiga, <a href="http://sequencedb.net/s/A237290">Computer-generated formulas for A237290</a>, Sequence Machine.
%F A237290 a(n) = A184387(n) - A237289(n).
%F A237290 a(p) = 2(p+2) for odd primes p.
%F A237290 a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
%F A237290 a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - _Antti Karttunen_, Nov 29 2024
%e A237290 For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
%e A237290 The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
%e A237290 a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.
%p A237290 isSumDist := proc(n,k)
%p A237290     local dvs,s ;
%p A237290     dvs := numtheory[divisors](n) ;
%p A237290     for s in combinat[powerset](dvs) do
%p A237290         add(m,m=op(s)) ;
%p A237290         if % = k then
%p A237290             return true;
%p A237290         end if;
%p A237290     end do:
%p A237290     false ;
%p A237290 end proc:
%p A237290 A237290 := proc(n)
%p A237290     local a;
%p A237290     a := 0 ;
%p A237290     for k from 1 to numtheory[sigma](n) do
%p A237290         if isSumDist(n,k) then
%p A237290             a := a+k;
%p A237290         end if;
%p A237290     end do:
%p A237290 end proc:
%p A237290 seq(A237290(n),n=1..20) ; # _R. J. Mathar_, Mar 13 2014
%t A237290 a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* _Giovanni Resta_, Mar 13 2014 *)
%o A237290 (PARI) padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
%o A237290 a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ _Michel Marcus_, Mar 09 2014
%o A237290 (PARI) A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after _Chai Wah Wu_'s Python-code, _Antti Karttunen_, Nov 29 2024
%o A237290 (Python)
%o A237290 from sympy import divisors
%o A237290 def A237290(n):
%o A237290     ds = divisors(n)
%o A237290     c, s = {0}, sum(ds)
%o A237290     for d in ds:
%o A237290         c |=  {a+d for a in c}
%o A237290     return sum(a for a in c if 1<=a<=s) # _Chai Wah Wu_, Jul 05 2023
%Y A237290 Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.
%K A237290 nonn
%O A237290 1,2
%A A237290 _Jaroslav Krizek_, Mar 02 2014