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A109883
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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.
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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) = p-1, a(p^n) = {p^(n+1) -2*p^n +1}/(p-1), if p is a prime.
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 1..10000
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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.
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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
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MATHEMATICA
| subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n] (from Bobby R. Treat, (DrBob(AT)bigfoot.com), Jul 14 2005)
Table[ f[n], {n, 80}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 14 2005)
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CROSSREFS
| Cf. A064510, A109884, A109886.
Sequence in context: A158032 A120112 A103977 * A033880 A033879 A033883
Adjacent sequences: A109880 A109881 A109882 * A109884 A109885 A109886
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KEYWORD
| easy,nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 11 2005
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EXTENSIONS
| More terms from Jason Earls (zevi_35711(AT)yahoo.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2005
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