

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.


10



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
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OFFSET

1,3


COMMENTS

If n is a perfect number then a(n) = 0. But if a(n) = 0, n need not be perfect, e.g., a(24) = 0, but 24 is not a perfect number. Also a(1) = 0, a(2^n) = 1, a(p) = p1, a(p^n) = {p^(n+1) 2*p^n +1}/(p1), if p is a prime.


LINKS

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


EXAMPLE

a(14) = 4: 141 = 13, 132 = 11, 117 = 4.
a(6) = 0: 61 =5, 52 = 3, 33 =0. 6 is a perfect number.
a(35) = 22: 351 = 34, 345=29, 297 = 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:=md[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]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n] (* Bobby R. Treat, (DrBob(AT)bigfoot.com), Jul 14 2005 *)
Table[ f[n], {n, 80}] (* Robert G. Wilson v, Jul 14 2005 *)


CROSSREFS

Cf. A064510, A109884, A109886.
Sequence in context: A120112 A233150 A103977 * A033880 A033879 A033883
Adjacent sequences: A109880 A109881 A109882 * A109884 A109885 A109886


KEYWORD

easy,nonn


AUTHOR

Amarnath Murthy, Jul 11 2005


EXTENSIONS

More terms from Jason Earls (zevi_35711(AT)yahoo.com) and Robert G. Wilson v, Jul 12 2005


STATUS

approved



