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A359196 a(n) is the number of subsets of the divisors of n which sum to n+1.

%I A359196 #32 Jan 20 2025 22:52:31
%S A359196 0,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,2,1,2,1,1,1,5,1,1,1,1,1,5,1,1,1,1,
%T A359196 1,8,1,1,1,3,1,3,1,1,1,1,1,11,1,1,1,1,1,3,1,3,1,1,1,33,1,1,1,1,1,3,1,
%U A359196 1,1,2,1,27,1,1,1,1,1,2,1,7,1,1,1,25,1,1,1,2,1,20,1,1,1,1,1,21,1,1,1,3
%N A359196 a(n) is the number of subsets of the divisors of n which sum to n+1.
%C A359196 Inspired by an email from Alonso Del Arte, dated Dec 12 2022.
%C A359196 Only abundant numbers, A005101, have values exceeding one.
%C A359196 First term to have k subsets is n = A359197(k).
%C A359196 Question: Are the positions of records given by some subset of A002182? See also A065218, A340840. - _Antti Karttunen_, Jan 20 2025
%H A359196 Antti Karttunen, <a href="/A359196/b359196.txt">Table of n, a(n) for n = 1..65537</a>
%e A359196 a(1) = 0; a(2) = 1 since the divisors of 2, {1, 2} sum to 3;
%e A359196 a(18) = 2 since the divisors of 18, {1, 2, 3, 6, 9, 18}, have two subsets, {1, 18}, {1, 3, 6, 9} which sum to 19;
%e A359196 a(12) = 3 since the divisors of 12, {1, 2, 3, 4, 6, 12}, have three subsets, {1, 12}, {3, 4, 6}, {1, 2, 4, 6} which sum to 13;
%e A359196 a(162) = 4 since its divisors are {1, 2, 3, 6, 9, 18, 27, 54, 81, 162}, have four subsets, {1, 162}, {1, 27, 54, 81}, {1, 9, 18, 54, 81}, {1, 3, 6, 18, 54, 81} which sum to 163;
%e A359196 a(24) = 5 since its divisors {1, 2, 3, 4, 6, 8, 12, 24} have five subsets {1, 24}, {1, 4, 8, 12}, {2, 3, 8, 12}, {3, 4, 6, 12}, {1, 2, 4, 6, 12} which sum to 25; etc.
%t A359196 a[n_] := Block[{c = k =1, d = Most@ Divisors@ n}, lgth = Length@ d; If[lgth < 18, c = 1 + Count[Total /@ Subsets@ d, n +1], While[k < 1 + 2^(lgth - 18), c += Count[Total /@ Subsets[d, All, {1 + (k -1)*2^18, k*2^18}], n +1]; k++]]; c]; Array[a, 100] (* or *)
%t A359196 a[n_] := Block[{d = Divisors@ n}, SeriesCoefficient[ Series[ Product[1 + x^d[[i]], {i, Length@ d}], {x, 0, n +1}], n +1]]; Array[a, 100]
%o A359196 (PARI) A359196(n) = if(!n, 0, if(sigma(n)<=n, 1, my(p=1); fordiv(n, d, p *= (1 + 'x^d)); polcoeff(p, 1+n))); \\ _Antti Karttunen_, Jan 20 2025
%Y A359196 Cf. A002182, A005101, A033630, A085491, A359197, A357050, A379504.
%K A359196 nonn
%O A359196 1,12
%A A359196 _Robert G. Wilson v_, Dec 19 2022