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

1,3

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.

Links

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

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

Example

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.

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(m<d[j+1])then return m:fi:od:end: # Nathaniel 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 *)

Prog

(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

Crossrefs

Cf. A064510, A109884, A109886, A117552.

Sequence in context: A120112 A233150 A103977 * A033880 A033879 A324546

Adjacent sequences: A109880 A109881 A109882 * A109884 A109885 A109886

Keyword

easy,nonn

Author

Amarnath Murthy, Jul 11 2005

Extensions

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

Status

approved