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A136446 Numbers n such that some subset of the numbers { 1 < d < n : d divides n } adds up to n. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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 *)

PROG

(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

(Haskell)

a136446 n = a136446_list !! (n-1)

a136446_list = map (+ 1) $ findIndices (> 1) a211111_list

-- Reinhard Zumkeller, Apr 04 2012

(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

CROSSREFS

See A005835 (allowing for divisor 1).

Cf. A122036 = A005231 \ A136446.

Sequence in context: A297925 A341099 A175837 * A074726 A341475 A091013

Adjacent sequences: A136443 A136444 A136445 * A136447 A136448 A136449

KEYWORD

nonn

AUTHOR

Joerg Arndt, Apr 06 2008

EXTENSIONS

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

STATUS

approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)