A136446 Numbers n such that some subset of the numbers { 1 < d < n : d divides n } adds up to n.

12, 18, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 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

Offset: 1

Author

Joerg Arndt, Apr 06 2008

Keywords

• nonn

Comments

This is a subset of the pseudoperfect numbers A005835 and thus non-deficient (A023196), but in view of the definition actually abundant numbers (A005101). Sequence A122036 lists odd abundant numbers (A005231) which are not in this sequence. So far, 351351 is the only one we know. (As of today, no odd weird (A006037: abundant but not pseudoperfect) number is known.) - M. F. Hasler, Apr 13 2008

This sequence contains infinitely many odd elements: any proper multiple of any pseudoperfect number is in the sequence, so odd proper multiples of odd pseudoperfect numbers are in the sequence. The first such is 2835 = 3 * 945 (which is in the b-file). - Franklin T. Adams-Watters, Jun 18 2009

A211111(a(n)) > 1. - Reinhard Zumkeller, Apr 04 2012

References

• Mladen Vassilev, Two theorems concerning divisors, Bull. Number Theory Related Topics 12 (1988), pp. 10-19.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..24491 (confirmed by _R. J. Mathar_, Mar 20 2011).

Crossrefs

See A005835 (allowing for divisor 1).

Cf. A122036 = A005231 \ A136446.

Programs

Haskell

a136446 n = a136446_list !! (n-1)
a136446_list = map (+ 1) $ findIndices (> 1) a211111_list
-- Reinhard Zumkeller, Apr 04 2012

Maple

isA136446a := proc(s,n) if n in s then return true; elif add(i,i=s) < n then return false; elif nops(s) = 1 then is(op(1,s)=n) ; else sl := sort(convert(s,list),`>`) ; for i from 1 to nops(sl) do m := op(i,sl) ; if n -m = 0 then return true; end if ; if n-m > 0 then sr := [op(i+1..nops(sl),sl)] ; if procname(convert(sr,set),n-m) then return true; end if; end if; end do; return false; end if; end proc:
isA136446 := proc(n) isA136446a( numtheory[divisors](n) minus {1,n},n) ; end proc:
for n from 1 to 400 do if isA136446(n) then printf("%d,",n) ; end if; end do ; # R. J. Mathar, Mar 20 2011

Mathematica

okQ[n_] := Module[{d}, If[PrimeQ[n], False, d = Most[Rest[Divisors[n]]]; MemberQ[Plus @@@ Subsets[d], n]]]; Select[Range[2, 246], okQ]
(* T. D. Noe, Jul 24 2012 *)

PARI

N=72 \\ up to this value
vv=vector(N);
{ for(n=2, N,
if ( isprime(n), next() );
d=divisors(n);
d=vector(#d-2,j,d[j+1]); \\ not n, not 1
for (k=1, (1<<#d)-1, \\ all subsets
t=vecextract(d, k);
if ( n==sum(j=1,#t,t[j]),
vv[n] += 1;););); }
for (j=1, #vv, if (vv[j]>0, print1(j,", "))) \\ A005835 (after correction)

PARI

is_A136446(n,d=divisors(n))={#d>2 && is_A005835(n,d[2..-2])} \\ Replaced old code not conforming to current PARI syntax. - M. F. Hasler, Jul 28 2016
for( n=1,10^4, is_A136446(n) && print1(n", ")) \\ M. F. Hasler, Apr 13 2008

Sage

def isa(s, n): # After R. J. Mathar's Maple code
    if n in s: return True
    if sum(s) < n: return False
    if len(s) == 1: return s[0] == n
    for i in srange(len(s)-1,-1,-1) :
        d = n - s[i]
        if d == 0: return True
        if d >  0:
            if isa(s[i+1:], d): return True
    return False
isA136446 = lambda n : isa(divisors(n)[1:-1], n)
[n for n in (1..246) if isA136446(n)]
# Peter Luschny, Jul 23 2012

Extensions

More terms from M. F. Hasler, Apr 13 2008