A225561 Largest number m such that 1, 2, ..., m can be represented as the sum of distinct divisors of n.

1, 3, 1, 7, 1, 12, 1, 15, 1, 3, 1, 28, 1, 3, 1, 31, 1, 39, 1, 42, 1, 3, 1, 60, 1, 3, 1, 56, 1, 72, 1, 63, 1, 3, 1, 91, 1, 3, 1, 90, 1, 96, 1, 7, 1, 3, 1, 124, 1, 3, 1, 7, 1, 120, 1, 120, 1, 3, 1, 168, 1, 3, 1, 127, 1, 144, 1, 7, 1, 3, 1, 195, 1, 3, 1, 7, 1, 168, 1, 186, 1, 3

Offset: 1

Author

Charles R Greathouse IV, May 10 2013

Keywords

• nonn
• look

Comments

n is called a practical number (A005153) if a(n) >= n.

Links

• Paul Tek, Table of n, a(n) for n = 1..10000
• Paul Pollack and Lola Thompson, Practical pretenders, Publicationes Mathematicae Debrecen, Vol. 82, No. 3-4 (2013), pp. 651-717, arXiv preprint, arXiv:1201.3168 [math.NT], 2012.
• Reinhard Zumkeller, Haskell programs for A201376, A054225, A201377, A054242

Crossrefs

Cf. A005153, A030057, A225574, A327832.

Cf. A033630, A201377, A027750.

Programs

Haskell

see Haskell link, 3.2.2
a225561 n = length $ takeWhile (not . null) $
            map (ps [] $ a027750_row n) [1..] where
   ps qs _      0  = [qs]
   ps   []       = []
   ps qs (k:ks) m  =
      if m == 0 then [] else ps (k:qs) ks (m - k) ++ ps qs ks m
-- Reinhard Zumkeller, May 11 2013

Mathematica

a[n_] := First[Complement[Range[DivisorSigma[1, n] + 1], Total /@ Subsets[Divisors[n]]]] - 1; Array[a, 100] (* Jean-François Alcover, Sep 27 2018 *)
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); g[n_] := If[(ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), ?(# > 1 &)]) == {}, n, Times @@ (Power @@@ fct[[1 ;; ind[[1, 1]] - 1]])]; a[n] := DivisorSigma[1, g[n]]; Array[a, 100] (* Amiram Eldar, Sep 27 2019 *)

PARI

a(n)=my(d=divisors(n),t,v=vector(2^#d-1,i,t=vecextract(d,i); sum(j=1,#t,t[j]))); v=vecsort(v,,8); for(i=1,#v,if(v[i]!=i,return(i-1)));v[#v]

Python

from sympy import divisors
def A225561(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    k = 1
    while k in c:
        k += 1
    return k-1 # Chai Wah Wu, Jul 05 2023

Formula

a(n) = 1 if and only if n is odd. a(n) = 3 if and only if n in {2,10} mod 12. Otherwise a(n) >= 7.

a(n) = A030057(n)-1.

a(n) = A000203(A327832(n)). - Amiram Eldar, Sep 27 2019