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A125225 Numbers n such that n-1 can be represented as a sum of a subset of divisors of n. 2
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The definition is related to that for semiperfect numbers (A005835). Every practical number (A005153) belongs to this sequence but not necessarily vice versa; e.g., 70 is in this sequence but not practical. Every number n in this sequence has sigma(n) >= 2n-1 (A103288) but, despite being abundant, 102 is not in this sequence.

Such numbers can be used to construct inheritance puzzles of the type described by Premchand Anne (see link).

Does the sequence contain A005231 (the odd abundant numbers)? - Robert Israel, Aug 05 2016

LINKS

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

Premchand Anne, Egyptian fractions and the inheritance problem, The College Mathematics Journal 29 (4) (1998) 296-300.

EXAMPLE

70 is in this sequence because 70-1=69=35+14+10+7+2+1 and all numbers in the sum are divisors of 70.

MAPLE

ss:= proc(n, S) local s, Sp;  option remember;

  if n = 0 then return true

  elif S = {} then return false

  fi;

  s:= max(S);

  if s > n then return procname(n, select(`<=`, S, n))

  elif s = n then return true

  fi;

  s:= min(S);

  Sp:= subs(s=NULL, S);

  if s > n then false

  else procname(n-s, Sp) or procname(n, Sp)

  fi

end proc:

select(n -> ss(n-1, numtheory:-divisors(n)), [$1..1000]); # Robert Israel, Aug 05 2016

PROG

(PARI) padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b); ); b; }

isok(n) = {if (n == 1, return (1)); d = divisors(n); nbd = #d; for (i = 1, 2^nbd-1, b = padbin(i, nbd); s = sum(j = 1, nbd, d[j]*b[j]); if (s == (n - 1), return (1)); ); return (0); } \\ Michel Marcus, Aug 30 2013

CROSSREFS

Cf. A005835, A000225, A005153, A103288.

Sequence in context: A301587 A319796 A103288 * A092903 A005153 A174973

Adjacent sequences:  A125222 A125223 A125224 * A125226 A125227 A125228

KEYWORD

nonn

AUTHOR

David Eppstein, Jan 13 2007

STATUS

approved

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Last modified June 1 04:57 EDT 2020. Contains 334758 sequences. (Running on oeis4.)