

A005835


Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.
(Formerly M4094)


34



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

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 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 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 a quarter. 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


REFERENCES

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).


LINKS

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


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); 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


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) 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 nM 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 !! (n1)
a005835_list = filter ((== 1) . a210455) [1..]
 Reinhard Zumkeller, Jan 21 2013


CROSSREFS

Subsequence of A023196; complement of 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


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Better description and more terms from Jud McCranie Oct 15 1997


STATUS

approved



