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A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

%I A119347 #52 Nov 29 2024 15:06:17
%S A119347 1,3,3,7,3,12,3,15,7,15,3,28,3,15,15,31,3,39,3,42,15,15,3,60,7,15,15,
%T A119347 56,3,72,3,63,15,15,15,91,3,15,15,90,3,96,3,63,55,15,3,124,7,63,15,63,
%U A119347 3,120,15,120,15,15,3,168,3,15,59,127,15,144,3,63,15,142,3,195,3,15,63,63
%N A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.
%C A119347 If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
%C A119347 The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - _T. D. Noe_, Mar 29 2010
%C A119347 The indices of records occur at the highly abundant numbers, excluding 3 and 10, if _Jaycob Coleman_'s conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - _Amiram Eldar_, Jun 13 2020
%C A119347 Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - _Antti Karttunen_ and _Ilya Gutkovskiy_, Nov 29 2024
%H A119347 Antti Karttunen, <a href="/A119347/b119347.txt">Table of n, a(n) for n = 1..20000</a> (first 10000 terms from T. D. Noe)
%H A119347 Jon Maiga, <a href="http://sequencedb.net/s/A119347">Computer-generated formulas for A119347</a>, Sequence Machine.
%H A119347 B. M. Stewart, <a href="https://doi.org/10.2307/2372651">Sums of distinct divisors</a>, American Journal of Mathematics 76 (1954), pp. 779-785.
%F A119347 For n > 1, 3 <= a(n) <= sigma(n). - _Charles R Greathouse IV_, Feb 11 2019
%F A119347 For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - _Ctibor O. Zizka_, Oct 19 2023
%F A119347 From _Antti Karttunen_, Nov 29 2024: (Start)
%F A119347 a(n) = A308605(n)-1.
%F A119347 a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
%F A119347 a(n) <= A100587(n).
%F A119347 (End)
%e A119347 a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.
%p A119347 with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);
%t A119347 a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
%t A119347 Array[a, 100] (* _Jean-François Alcover_, Jan 27 2018 *)
%o A119347 (Haskell)
%o A119347 import Data.List (subsequences, nub)
%o A119347 a119347 = length . nub . map sum . tail . subsequences . a027750_row'
%o A119347 -- _Reinhard Zumkeller_, Jun 27 2015
%o A119347 (Python)
%o A119347 from sympy import divisors
%o A119347 def A119347(n):
%o A119347     c = {0}
%o A119347     for d in divisors(n,generator=True):
%o A119347         c |=  {a+d for a in c}
%o A119347     return len(c)-1 # _Chai Wah Wu_, Jul 05 2023
%o A119347 (PARI) A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0<polcoeff(p, i))); }; \\ _Antti Karttunen_, Nov 28 2024
%o A119347 (PARI) A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after _Chai Wah Wu_'s Python-code, _Antti Karttunen_, Nov 29 2024
%Y A119347 One less than A308605.
%Y A119347 Cf. A000203, A002093, A005153, A027750, A030057, A033630, A093890, A100587, A119348, A193279, A225561, A237290, A378447, A378450, A378451.
%Y A119347 Cf. A083207 (positions of even terms).
%K A119347 nonn
%O A119347 1,2
%A A119347 _Emeric Deutsch_, May 15 2006
%E A119347 Definition clarified by _Antti Karttunen_, Nov 29 2024