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
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
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.
Comments