This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193279 #34 Nov 29 2024 15:10:16
%S A193279 0,1,1,3,1,6,1,7,3,7,1,16,1,7,7,15,1,21,1,22,7,7,1,36,3,7,7,28,1,42,1,
%T A193279 31,7,7,7,55,1,7,7,50,1,54,1,31,27,7,1,76,3,31,7,31,1,66,7,64,7,7,1,
%U A193279 108,1,7,29,63,7,78,1,31,7,72,1,123,1,7,31,31
%N A193279 Number of distinct sums of distinct proper divisors of n.
%C A193279 a(n)=1 if and only if n is prime.
%C A193279 a(n)=n-1 if n is a power of 2.
%C A193279 a(n)=n if n is an even perfect number (is the converse true?)
%C A193279 Note: the count excludes an empty subset of proper divisors that would give 0 as a sum. - _Antti Karttunen_, Mar 07 2018
%H A193279 Antti Karttunen, <a href="/A193279/b193279.txt">Table of n, a(n) for n = 1..20000</a> (first 10000 terms from Amiram Eldar)
%p A193279 with(linalg): a:=proc(n) local dl,t: dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): return nops({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)}): end: seq(a(n), n=1..76); # _Nathaniel Johnston_, Jul 23 2011
%t A193279 a[n_] := Module[{d = Most @ Divisors[n], x}, Count[CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] - 1]; Array[a, 100] (* _Amiram Eldar_, Jun 13 2020 *)
%o A193279 (PARI)
%o A193279 \\ Slow and naive:
%o A193279 A193279(n) = if(1==n,0,my(pds = (divisors(n)[1..(numdiv(n)-1)]), maxsum = vecsum(pds), sums = vector(maxsum), psetsiz = (2^length(pds))-1, k = 0, s); for(i=1,psetsiz,s = vecsum(choosebybits(pds,i)); if(!sums[s],k++;sums[s]++)); (k)); \\ _Antti Karttunen_, Mar 07 2018
%o A193279 (PARI) A193279(n) = { my(p=1); fordiv(n, d, if(d<n, p *= (1 + 'x^d))); sum(i=1,poldegree(p),(0<polcoeff(p, i))); }; \\ _Antti Karttunen_, Nov 29 2024
%o A193279 (PARI) A193279(n) = { my(c=[0]); fordiv(n,d, if(d<n, c = Set(concat(c,vector(#c,i,c[i]+d))))); (#c)-1; }; \\ (see A119347) - _Antti Karttunen_, Nov 29 2024
%Y A193279 Cf. A193280.
%Y A193279 Cf. A119347 (allows also n to be included in the sums), A378447 (differences).
%K A193279 nonn
%O A193279 1,4
%A A193279 _Michael Engling_, Jul 20 2011