cp's OEIS Frontend

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

%I A030057 #62 Aug 22 2024 03:57:45
%S A030057 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 A030057 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 A030057 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 A030057 Least number that is not a sum of distinct divisors of n.
%C A030057 a(n) = 2 if and only if n is odd. a(2^n) = 2^(n+1). - _Emeric Deutsch_, Aug 07 2005
%C A030057 a(n) > n if and only if n belongs to A005153, and then a(n) = sigma(n) + 1. - _Michel Marcus_, Oct 18 2013
%C A030057 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 A030057 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 A030057 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 A030057 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 A030057 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 A030057 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 A030057 (Haskell)
%o A030057 a030057 n = head $ filter ((== 0) . p (a027750_row n)) [1..] where
%o A030057    p _      0 = 1
%o A030057    p []     _ = 0
%o A030057    p (k:ks) x = if x < k then 0 else p ks (x - k) + p ks x
%o A030057 -- _Reinhard Zumkeller_, Feb 27 2012
%o A030057 (Python)
%o A030057 from sympy import divisors
%o A030057 def A030057(n):
%o A030057     c = {0}
%o A030057     for d in divisors(n,generator=True):
%o A030057         c |=  {a+d for a in c}
%o A030057     k = 1
%o A030057     while k in c:
%o A030057         k += 1
%o A030057     return k # _Chai Wah Wu_, Jul 05 2023
%Y A030057 Cf. A002093, A005153, A093896, A119347.
%Y A030057 Distinct elements form A030058.
%Y A030057 Cf. A027750.
%K A030057 nonn,nice,look
%O A030057 1,1
%A A030057 _David W. Wilson_
%E A030057 Edited by _N. J. A. Sloane_, May 05 2007