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)
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
REFERENCES
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).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Anonymous, Semiperfect Numbers: Definition [Broken link]
Stan Benkoski, Problem E2308, Amer. Math. Monthly, Vol. 79, No. 7 (1972), p. 774.
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., Vol. 28, No. 126 (1974), pp. 617-623. Corrigendum, Math. Comp., Vol. 29, No. 130 (1975), pp. 673-674.
David Eppstein, Is it known whether a group of Egyptian fractions with odd, distinct denominators can add up to 1?, 1996.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
Wacław Sierpiński, Sur les nombres pseudoparfaits, Matematički Vesnik, Vol. 2 (17), No. 33 (1965), pp. 212-213.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; arXiv:math preprint, arXiv:math/1812.06566 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Semiperfect Number.
Wikipedia, Semiperfect number.
Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.
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
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 *)
PROG
(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
for(n=1, 1000, is_A005835(n)&&print1(n", ")) \\ M. F. Hasler, Apr 06 2008
(Haskell)
a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
-- Reinhard Zumkeller, Jan 21 2013
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Better description and more terms from Jud McCranie, Oct 15 1997
STATUS
approved