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A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

%I A378450 #21 Nov 29 2024 12:23:02
%S A378450 0,0,1,0,3,0,5,0,6,3,9,0,11,9,9,0,15,0,17,0,17,21,21,0,24,27,25,0,27,
%T A378450 0,29,0,33,39,33,0,35,45,41,0,39,0,41,21,23,57,45,0,50,30,57,35,51,0,
%U A378450 57,0,65,75,57,0,59,81,45,0,69,0,65,63,81,2,69,0,71,99,61,77,81,0,77,0,90,111,81,0,93,117,105
%N A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.
%C A378450 Difference between the sum of divisors n and the number of distinct sums of distinct divisors of n.
%H A378450 Antti Karttunen, <a href="/A378450/b378450.txt">Table of n, a(n) for n = 1..20000</a>
%F A378450 a(n) = A000203(n) - A119347(n).
%e A378450 For n = 3, with divisors [1, 3] and sigma(3)=4, only 2 in range 1..4 cannot be represented as a sum of a subset of [1, 3], therefore a(3) = 1.
%e A378450 For n = 15, with divisors [1, 3, 5, 15] and sigma(15) = 24, the subset sums are 1, 3, 1+3, 5, 1+5, 3+5, 1+3+5, 15, 1+15, 3+15, 1+3+15, 5+15, 1+5+15, 3+5+15, 1+3+5+15 i.e., [1, 3, 4, 5, 6, 8, 9, 15, 16, 18, 19, 20, 21, 23, 24], which leaves 2, 7, 10, 11, 12, 13, 14, 17, 22 as unrepresented numbers, therefore a(15) = 9.
%o A378450 (PARI)
%o A378450 A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
%o A378450 A378450(n) = (sigma(n)-A119347(n));
%Y A378450 Cf. A000203, A119347, A237289 (gives the sums of unrepresented numbers), A322860.
%Y A378450 Cf. A005153 (positions of 0's), A237287 (of nonzeros), A030057.
%K A378450 nonn
%O A378450 1,5
%A A378450 _Antti Karttunen_, Nov 29 2024