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

1,2

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

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

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 *)

Prog

(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]
(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
(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

Crossrefs

Cf. A005153, A030057, A225574, A327832.

Cf. A033630, A201377, A027750.

Sequence in context: A266724 A247146 A362628 * A098093 A160627 A114712

Adjacent sequences: A225558 A225559 A225560 * A225562 A225563 A225564

Keyword

nonn,look

Author

Charles R Greathouse IV, May 10 2013

Status

approved