A109883 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.

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, 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, 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

Offset: 1

Author

Amarnath Murthy, Jul 11 2005

Keywords

• easy
• nonn

Comments

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.

Examples

			a(14) = 4: 14-1 = 13, 13-2 = 11, 11-7 = 4.
a(6) = 0: 6-1 = 5, 5-2 = 3, 3-3 = 0. 6 is a perfect number.
a(35) = 22: 35-1 = 34, 34-5 = 29, 29-7 = 22.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A064510, A109884, A109886, A117552.

Programs

Maple

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(mNathaniel Johnston, Apr 15 2011

Mathematica

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&;
a[n_] := Catch @ Fold[subtract, n, Divisors @ n]
Table[ a[n], {n, 80}] (* Bobby R. Treat (DrBob(AT)bigfoot.com), Jul 14 2005 *)

PARI

a(n) = {my(r = n); fordiv(n, d, if (r < d, return (r)); r -= d;); 0;} \\ Michel Marcus, Dec 28 2018

Python

from sympy import divisors
def A109883(n):
    if n == 1: return 0
    s = n
    for d in divisors(n)[:-1]:
        if s < d: break
        s -= d
    return s
print([A109883(n) for n in range(1, 80)]) # Michael S. Branicky, Mar 31 2024

Formula

a(1) = 0, a(2^n) = 1.

a(p) = p-1, a(p^n) = (p^(n+1) - 2*p^n + 1)/(p-1), if p is a prime.

a(n) = n - A117552(n). - Ridouane Oudra, Jan 25 2024

Extensions

More terms from Jason Earls and Robert G. Wilson v, Jul 12 2005