1,1

In other words, some subset of the numbers { 1 <= d < n : d divides n } adds up to n. - N. J. A. Sloane (njas(AT)research.att.com), Apr 06 2008

Also, numbers n such that A033630(n) > 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 02 2007

By definition, does not include the weird numbers A006037.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Problem E2308, Amer. Math. Monthly, 79 (1972), 774.

T. D. Noe, Table of n, a(n) for n=1..1000

Anonymous, Semiperfect Numbers: Definition

David Eppstein, Title?

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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.

with(combinat); issemiperfect := proc(n) local b, S; b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if convert(S[nextvalue](), `+`)=n then b:=true; break fi od; return b end: select(proc(z) issemiperfect(z) end, [$1..1000]); - Walter A. Kehowski (wkehowski(AT)cox.net), Aug 12 2005

Flatten[ Position[ A033630, q_/; q>1 ] ] - from wouter.meeussen(at)pandora.be

(PARI from M. F. hasler, Apr 06 2008) isA005835(n, d=0)={ local(t); /* Return nonzero iff n is the sum of a subset of d which defaults to the set of proper divisors of n */

if( !d, /* Initialize d */ d=vecextract(divisors(n), "^-1"), /*else check if n equals one element of d */ setsearch( Set(d), n) & return(1));

/* Remove terms > n */ while( #d>1 & d[ #d]>n, d=vecextract(d, "^-1"));

/* If n is not smaller than the sum of all terms, we're done */ n >= (t = sum(i=1, #d, d[i])) & return( n==t );

/* If n is larger than M=max(d), then try to write n-M in terms of d \ { M } */ n > d[ #d ] & isA005835( n - d[ #d ], vecextract( d, "^-1") ) & return(1); /* else only d \ {M} is needed */ isA005835( n, vecextract( d, "^-1" ))}

for(n=1, 1000, isA005835(n)&print1(n", "))

The complement is A136447.

See A136446 for another version.

Cf. A006036, A005100, A033630.

Sequence in context: A119357 A097216 A023196 this_sequence A007620 A100715 A094519

Adjacent sequences: A005832 A005833 A005834 this_sequence A005836 A005837 A005838

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Better description and more terms from Jud McCranie (j.mccranie(AT)comcast.net) Oct 15 1997