A030057 - OEIS

%I #62 Aug 22 2024 03:57:45

%S 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,

%T 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,

%U 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

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

%C a(n) = 2 if and only if n is odd. a(2^n) = 2^(n+1). - _Emeric Deutsch_, Aug 07 2005

%C a(n) > n if and only if n belongs to A005153, and then a(n) = sigma(n) + 1. - _Michel Marcus_, Oct 18 2013

%C 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

%C 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

%H David Wasserman and T. D. Noe, <a href="/A030057/b030057.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from David Wasserman)

%e 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.

%p 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

%t 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 *)

%o (Haskell)

%o a030057 n = head $ filter ((== 0) . p (a027750_row n)) [1..] where

%o    p _      0 = 1

%o    p []     _ = 0

%o    p (k:ks) x = if x < k then 0 else p ks (x - k) + p ks x

%o -- _Reinhard Zumkeller_, Feb 27 2012

%o (Python)

%o from sympy import divisors

%o def A030057(n):

%o     c = {0}

%o     for d in divisors(n,generator=True):

%o         c |=  {a+d for a in c}

%o     k = 1

%o     while k in c:

%o         k += 1

%o     return k # _Chai Wah Wu_, Jul 05 2023

%Y Cf. A002093, A005153, A093896, A119347.

%Y Distinct elements form A030058.

%Y Cf. A027750.

%K nonn,nice,look

%O 1,1

%A _David W. Wilson_

%E Edited by _N. J. A. Sloane_, May 05 2007