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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

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

Examples

			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.
		

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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.

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.

Crossrefs

Subsequence of A023196; complement of A136447.

See A136446 for another version.

Cf. A006036, A005100, A033630, A000396, A005231.

Cf. A109761 (subsequence).

Programs

Haskell

a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
-- Reinhard Zumkeller, Jan 21 2013

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 ] ] (* 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 *)

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(nA005835(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

Extensions

Better description and more terms from Jud McCranie, Oct 15 1997