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A225561 Largest number m such that 1, 2, ..., m can be represented as the sum of distinct divisors of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

CROSSREFS

Cf. A005153, A030057, A225574, A327832.

Cf. A033630, A201377, A027750.

Sequence in context: A050521 A266724 A247146 * A098093 A160627 A114712

Adjacent sequences:  A225558 A225559 A225560 * A225562 A225563 A225564

KEYWORD

nonn,look

AUTHOR

Charles R Greathouse IV, May 10 2013

STATUS

approved

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Last modified October 21 16:25 EDT 2019. Contains 328302 sequences. (Running on oeis4.)