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A136446
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Numbers n such that some subset of the numbers { 1 < d < n : d divides n } adds up to n.
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5
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| This is a subset of the pseudoperfect numbers A005835 and thus abundant numbers A005101. Sequence A122036 lists odd abundant numbers (A005231) which are not in this sequence. (As of today, no odd abundant number is known which is not pseudoperfect.) - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 13 2008
Values up to a(24491) confirmed - R. J. Mathar Mar 20 2011
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). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 18 2009]
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LINKS
| M. F. Hasler, Table of n, a(n) for n=1,...,24491.
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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
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PROG
| (PARI/gp) 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 code from M. F. Hasler, Apr 13 2008) /* this is equivalent to sigma(n)>2*n & isA005835(n, vecextract(divisors(n), "2..-2")) */
isA136446(n, d=0)={ local(t); if( !d, sigma(n)>2*n | return; d=vecextract(divisors(n), "2..-2"), setsearch( Set(d), n)&return(1)); while(#d>1&d[ #d ]>n, d=vecextract(d, "^-1")); n>=(t=sum(i=1, #d, d[ i ])) & return(n==t); n>d[ #d ] & isA136446(n-d[ #d ], vecextract(d, "^-1")) & return(1); isA136446(n, vecextract(d, "^-1"))}
for( n=1, 10^4, isA136446(n) & print1(n", "))
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CROSSREFS
| See A005835 (allowing for divisor 1).
Cf. A122036 = A005231 \ A136446.
Sequence in context: A162151 A056773 A175837 * A074726 A091013 A159886
Adjacent sequences: A136443 A136444 A136445 * A136447 A136448 A136449
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KEYWORD
| nonn
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AUTHOR
| Joerg Arndt (arndt(AT)jjj.de), Apr 06 2008
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EXTENSIONS
| More terms from M. F. Hasler, Apr 13 2008
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