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A005835
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Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.
(Formerly M4094)
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70
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6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264
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OFFSET
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1,1
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COMMENTS
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In other words, some subset of the numbers { 1 <= d < n : d divides n } adds up to n. - N. J. A. Sloane, Apr 06 2008
Deficient numbers cannot be pseudoperfect. This sequence includes the perfect numbers (A000396). By definition, it does not include the weird, i.e., abundant but not pseudoperfect, numbers (A006037).
The first odd pseudoperfect number is a(233) = 945.
An empirical observation (from the graph) is that it seems that the n-th pseudoperfect number would be asymptotic to 4n, or equivalently that the asymptotic density of pseudoperfect numbers would be 1/4. Any proof of this? (End)
Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790. - Jaycob Coleman, Oct 26 2013
The odd terms of this sequence are given by the odd abundant numbers A005231, up to hypothetical (so far unknown) odd weird numbers (A006037). - M. F. Hasler, Nov 23 2017
The term "pseudoperfect numbers" was coined by Sierpiński (1965). The alternative term "semiperfect numbers" was coined by Zachariou and Zachariou (1972). - Amiram Eldar, Dec 04 2020
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Stan Benkoski, Problem E2308, Amer. Math. Monthly, Vol. 79, No. 7 (1972), p. 774.
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EXAMPLE
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6 = 1+2+3, 12 = 1+2+3+6, 18 = 3+6+9, etc.
70 is not a member since the proper divisors of 70 are {1, 2, 5, 7, 10, 14, 35} and no subset adds to 70.
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MAPLE
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with(combinat):
isA005835 := proc(n)
local b, S;
b:=false;
S:=subsets(numtheory[divisors](n) minus {n});
while not S[finished] do
if convert(S[nextvalue](), `+`)=n then
b:=true;
break
end if ;
end do;
b
end proc:
for n from 1 do
if isA005835(n) then
print(n);
end if;
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MATHEMATICA
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pseudoPerfectQ[n_] := Module[{divs = Most[Divisors[n]]}, MemberQ[Total/@Subsets[ divs, Length[ divs]], n]]; A005835 = Select[Range[300], pseudoPerfectQ] (* Harvey P. Dale, Sep 19 2011 *)
A005835 = {}; n = 0; While[Length[A005835] < 100, n++; d = Most[Divisors[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[A005835, n]]]; A005835 (* T. D. Noe, Dec 29 2011 *)
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PROG
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(PARI) is_A005835(n, d=divisors(n)[^-1], s=vecsum(d), m=#d)={ m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(n<s, is_A005835(n-d[m], d, s-d[m], m-1) || is_A005835(n, d, s-d[m], m-1), n==s)} \\ Returns nonzero iff n is the sum of a subset of d, which defaults to the set of proper divisors of n. Improved using more recent PARI syntax by M. F. Hasler, Jul 15 2016, Jul 27 2016. NOTE: This function is also used (with 2nd optional arg) in A136446, A122036 and possibly in A006037. - M. F. Hasler, Jul 28 2016
(Haskell)
a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Better description and more terms from Jud McCranie, Oct 15 1997
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STATUS
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approved
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