A030057 Least number that is not a sum of distinct divisors of n.

2, 4, 2, 8, 2, 13, 2, 16, 2, 4, 2, 29, 2, 4, 2, 32, 2, 40, 2, 43, 2, 4, 2, 61, 2, 4, 2, 57, 2, 73, 2, 64, 2, 4, 2, 92, 2, 4, 2, 91, 2, 97, 2, 8, 2, 4, 2, 125, 2, 4, 2, 8, 2, 121, 2, 121, 2, 4, 2, 169, 2, 4, 2, 128, 2, 145, 2, 8, 2, 4, 2, 196, 2, 4, 2, 8, 2, 169, 2, 187, 2, 4, 2, 225, 2, 4, 2, 181

Offset

1,1

Comments

a(n) = 2 if and only if n is odd. a(2^n) = 2^(n+1). - Emeric Deutsch, Aug 07 2005

a(n) > n if and only if n belongs to A005153, and then a(n) = sigma(n) + 1. - Michel Marcus, Oct 18 2013

The most frequent values are 2 (50%), 4 (16.7%), 8 (5.7%), 13 (3.2%), 16 (2.4%), 29 (1.3%), 32 (1%), 40, 43, 61, ... - M. F. Hasler, Apr 06 2014

The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020

Links

David Wasserman and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from David Wasserman)

Example

a(10)=4 because 4 is the least positive integer that is not a sum of distinct divisors (namely 1,2,5 and 10) of 10.

Maple

with(combinat): with(numtheory): for n from 1 to 100 do div:=powerset(divisors(n)): b[n]:=sort({seq(sum(div[i][j], j=1..nops(div[i])), i=1..nops(div))}) od: for n from 1 to 100 do B[n]:={seq(k, k=0..1+sigma(n))} minus b[n] od: seq(B[n][1], n=1..100); # Emeric Deutsch, Aug 07 2005

Mathematica

a[n_] :=  First[ Complement[ Range[ DivisorSigma[1, n] + 1], Total /@ Subsets[ Divisors[n]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 02 2012 *)

Prog

(Haskell)
a030057 n = head $ filter ((== 0) . p (a027750_row n)) [1..] where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) x = if x < k then 0 else p ks (x - k) + p ks x
-- Reinhard Zumkeller, Feb 27 2012
(Python)
from sympy import divisors
def A030057(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 # Chai Wah Wu, Jul 05 2023

Crossrefs

Cf. A002093, A005153, A093896, A119347.

Distinct elements form A030058.

Cf. A027750.

Sequence in context: A059866 A278262 A093895 * A286596 A134066 A090988

Adjacent sequences: A030054 A030055 A030056 * A030058 A030059 A030060

Keyword

nonn,nice,look

Author

David W. Wilson

Extensions

Edited by N. J. A. Sloane, May 05 2007

Status

approved