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A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.

%I A005835 M4094 #134 Jul 18 2025 10:33:15
%S A005835 6,12,18,20,24,28,30,36,40,42,48,54,56,60,66,72,78,80,84,88,90,96,100,
%T A005835 102,104,108,112,114,120,126,132,138,140,144,150,156,160,162,168,174,
%U A005835 176,180,186,192,196,198,200,204,208,210,216,220,222,224,228,234,240,246,252,258,260,264
%N A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.
%C A005835 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
%C A005835 Also, numbers n such that A033630(n) > 1. - _Reinhard Zumkeller_, Mar 02 2007
%C A005835 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).
%C A005835 From _Daniel Forgues_, Feb 07 2011: (Start)
%C A005835 The first odd pseudoperfect number is a(233) = 945.
%C A005835 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)
%C A005835 A065205(a(n)) > 0; A210455(a(n)) = 1. - _Reinhard Zumkeller_, Jan 21 2013
%C A005835 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
%C A005835 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
%C A005835 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
%D A005835 Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
%D A005835 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005835 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.
%H A005835 Amiram Eldar, <a href="/A005835/b005835.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A005835 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node36.html">Semiperfect Numbers: Definition</a> [Broken link]
%H A005835 Stan Benkoski, <a href="http://dx.doi.org/10.2307%2F2316276">Problem E2308</a>, Amer. Math. Monthly, Vol. 79, No. 7 (1972), p. 774.
%H A005835 S. J. Benkoski and P. Erdős, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0347726-9">On weird and pseudoperfect numbers</a>, Math. Comp., Vol. 28, No. 126 (1974), pp. 617-623. <a href="https://doi.org/10.1090/S0025-5718-75-99676-3">Corrigendum</a>, Math. Comp., Vol. 29, No. 130 (1975), pp. 673-674.
%H A005835 David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html">Is it known whether a group of Egyptian fractions with odd, distinct denominators can add up to 1?</a>, 1996.
%H A005835 Richard K. Guy, <a href="https://doi.org/10.1007/978-0-387-26677-0">Unsolved Problems in Number Theory</a>, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
%H A005835 Tyler Ross, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Ross/ross3.html">A Perfect Number Generalization and Some Euclid-Euler Type Results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
%H A005835 Wacław Sierpiński, <a href="https://eudml.org/doc/259169">Sur les nombres pseudoparfaits</a>, Matematički Vesnik, Vol. 2 (17), No. 33 (1965), pp. 212-213.
%H A005835 Jonathan Sondow and Kieren MacMillan, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.3.232">Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation</a>, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; <a href="http://arxiv.org/abs/1812.06566">arXiv:math preprint</a>, arXiv:math/1812.06566 [math.NT], 2018.
%H A005835 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SemiperfectNumber.html">Semiperfect Number</a>.
%H A005835 Wikipedia, <a href="http://en.wikipedia.org/wiki/Semiperfect_number">Semiperfect number</a>.
%H A005835 Andreas Zachariou and Eleni Zachariou, <a href="http://www.hms.gr/apothema/?s=sap&amp;i=261">Perfect, Semi-Perfect and Ore Numbers</a>, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; <a href="https://eudml.org/doc/238923">alternative link</a>.
%e A005835 6 = 1+2+3, 12 = 1+2+3+6, 18 = 3+6+9, etc.
%e A005835 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.
%p A005835 with(combinat):
%p A005835 isA005835 := proc(n)
%p A005835     local b, S;
%p A005835     b:=false;
%p A005835     S:=subsets(numtheory[divisors](n) minus {n});
%p A005835     while not S[finished] do
%p A005835         if convert(S[nextvalue](), `+`)=n then
%p A005835             b:=true;
%p A005835             break
%p A005835         end if ;
%p A005835     end do;
%p A005835     b
%p A005835 end proc:
%p A005835 for n from 1 do
%p A005835     if isA005835(n) then
%p A005835         print(n);
%p A005835     end if;
%p A005835 end do: # _Walter Kehowski_, Aug 12 2005
%t A005835 A005835 = Flatten[ Position[ A033630, q_/; q>1 ] ] (* _Wouter Meeussen_ *)
%t A005835 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 *)
%t A005835 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 *)
%o A005835 (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
%o A005835 for(n=1,1000,is_A005835(n)&&print1(n",")) \\ _M. F. Hasler_, Apr 06 2008
%o A005835 (Haskell)
%o A005835 a005835 n = a005835_list !! (n-1)
%o A005835 a005835_list = filter ((== 1) . a210455) [1..]
%o A005835 -- _Reinhard Zumkeller_, Jan 21 2013
%Y A005835 Subsequence of A023196; complement of A136447.
%Y A005835 See A136446 for another version.
%Y A005835 Cf. A006036, A005100, A033630, A000396, A005231.
%Y A005835 Cf. A109761 (subsequence).
%K A005835 nonn,nice,easy
%O A005835 1,1
%A A005835 _N. J. A. Sloane_
%E A005835 Better description and more terms from _Jud McCranie_, Oct 15 1997