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%I #28 Apr 01 2024 02:33:31
%S 0,1,2,1,4,0,6,1,5,2,10,2,12,4,6,1,16,6,18,8,10,8,22,0,19,10,14,0,28,
%T 3,30,1,18,14,22,11,36,16,22,10,40,9,42,4,12,20,46,12,41,7,30,6,52,15,
%U 38,20,34,26,58,2,60,28,22,1,46,21,66,10,42,31,70,9,72,34,26,12,58,27,78
%N Start subtracting from n its divisors beginning from 1 until one reaches a number smaller than the last divisor subtracted or reaches the last nontrivial divisor < n. Define this to be the perfect deficiency of n. Then a(n) = perfect deficiency of n.
%C If n is a perfect number then a(n) = 0. But if a(n) = 0, n needs not be perfect, e.g., a(24) = 0, but 24 is not a perfect number. See A064510.
%H Nathaniel Johnston, <a href="/A109883/b109883.txt">Table of n, a(n) for n = 1..10000</a>
%F a(1) = 0, a(2^n) = 1.
%F a(p) = p-1, a(p^n) = (p^(n+1) - 2*p^n + 1)/(p-1), if p is a prime.
%F a(n) = n - A117552(n). - _Ridouane Oudra_, Jan 25 2024
%e a(14) = 4: 14-1 = 13, 13-2 = 11, 11-7 = 4.
%e a(6) = 0: 6-1 = 5, 5-2 = 3, 3-3 = 0. 6 is a perfect number.
%e a(35) = 22: 35-1 = 34, 34-5 = 29, 29-7 = 22.
%p A109883:=proc(n)local d,j,k,m:if(n=1)then return 0:fi:j:=1:m:=n:d:=divisors(n);k:=nops(d):for j from 1 to k do m:=m-d[j]:if(m<d[j+1])then return m:fi:od:end: # _Nathaniel Johnston_, Apr 15 2011
%t subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&;
%t a[n_] := Catch @ Fold[subtract, n, Divisors @ n]
%t Table[ a[n], {n, 80}] (* Bobby R. Treat (DrBob(AT)bigfoot.com), Jul 14 2005 *)
%o (PARI) a(n) = {my(r = n); fordiv(n, d, if (r < d, return (r)); r -= d;); 0;} \\ _Michel Marcus_, Dec 28 2018
%o (Python)
%o from sympy import divisors
%o def A109883(n):
%o if n == 1: return 0
%o s = n
%o for d in divisors(n)[:-1]:
%o if s < d: break
%o s -= d
%o return s
%o print([A109883(n) for n in range(1, 80)]) # _Michael S. Branicky_, Mar 31 2024
%Y Cf. A064510, A109884, A109886, A117552.
%K easy,nonn
%O 1,3
%A _Amarnath Murthy_, Jul 11 2005
%E More terms from _Jason Earls_ and _Robert G. Wilson v_, Jul 12 2005
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