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 A005835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n. (Formerly M4094) 42
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 Also, numbers n such that A033630(n) > 1. - Reinhard Zumkeller, Mar 02 2007 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). From Daniel Forgues, Feb 07 2011: (Start) 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) A065205(a(n)) > 0; A210455(a(n)) = 1. - Reinhard Zumkeller, Jan 21 2013 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 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, Section B2. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Anonymous, Semiperfect Numbers: Definition [Broken link] Stan Benkoski, Problem E2308, Amer. Math. Monthly, 79 (1972), 774. S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617-623. Corrigendum: Math. Comp. 29 (1975), 673-674. R. K. Guy, Unsolved Problems in Number Theory, B2. J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240. Eric Weisstein's World of Mathematics, Semiperfect Number Wikipedia, Semiperfect number EXAMPLE 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. MAPLE 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; end do: # Walter Kehowski, Aug 12 2005 MATHEMATICA A005835 = Flatten[ Position[ A033630, q_/; q>1 ] ] (* from Wouter Meeussen *) 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 *) PROG (PARI) is_A005835(n, d=divisors(n)[^-1], s=vecsum(d), m=#d)={ m||return; while(d[m]>n, s-=d[m]; m--); d[m]==n || if(n

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