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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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33
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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
Also, numbers n such that A033630(n) > 1. - Reinhard Zumkeller, Mar 02 2007
Deficient numbers can not be pseudoperfect. This sequence includes the perfect numbers (A000396). By definition, it does not include the weird, i.e. abundant while not pseudoperfect, numbers (A006037).
[Daniel Forgues, Feb 7 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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B2.
Problem E2308, Amer. Math. Monthly, 79 (1972), 774.
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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T. D. Noe, Table of n, a(n) for n=1..1000
Anonymous, Semiperfect Numbers: Definition
David Eppstein, Is it known whether a group of Egyptian fractions with odd, distinct denominators can add up to 1?
Eric Weisstein's World of Mathematics, Semiperfect Number
Wikipedia, Semiperfect number
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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); issemiperfect := proc(n) local b, S; b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if convert(S[nextvalue](), `+`)=n then b:=true; break fi od; return b end: select(proc(z) issemiperfect(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005
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MATHEMATICA
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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] (* From 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) isA005835(n, d=0)={ local(t); /* Return nonzero iff n is the sum of a subset of d which defaults to the set of proper divisors of n */
if( !d, /* Initialize d */ d=vecextract(divisors(n), "^-1"), /*else check if n equals one element of d */ setsearch( Set(d), n) & return(1));
/* Remove terms > n */ while( #d>1 & d[ #d]>n, d=vecextract(d, "^-1"));
/* If n is not smaller than the sum of all terms, we're done */ n >= (t = sum(i=1, #d, d[i])) & return( n==t );
/* If n is larger than M=max(d), then try to write n-M in terms of d \ { M } */ n > d[ #d ] & isA005835( n - d[ #d ], vecextract( d, "^-1") ) & return(1); /* else only d \ {M} is needed */ isA005835( n, vecextract( d, "^-1" ))}
for(n=1, 1000, isA005835(n)&print1(n", ")) /* from M. F. hasler, Apr 06 2008 */
(Haskell)
a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
-- Reinhard Zumkeller, Jan 21 2013
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CROSSREFS
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The complement is A136447.
See A136446 for another version.
Cf. A006036, A005100, A033630, A000396.
Cf. A109761 (subsequence).
Sequence in context: A177052 A023196 A204829 * A007620 A100715 A205525
Adjacent sequences: A005832 A005833 A005834 * A005836 A005837 A005838
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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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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