A058971 For a rational number p/q let f(p/q) = sum of divisors of p+q divided by number of divisors of p+q; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.
3, 2, 6, 3, 3, 4, 10, 87, 6, 6, 9, 7, 6, 6, 87, 9, 6, 10, 7, 8, 9, 12, 9, 15, 12, 10, 16, 15, 9, 16, 12, 12, 15, 12, 87, 19, 15, 14, 19, 21, 12, 22, 14, 13, 18, 24, 34, 19, 12, 18, 0, 27, 15, 18, 15, 20, 24, 30, 14, 31, 24, 18, 51, 21, 18, 34, 21, 24, 18, 36, 24, 37, 30, 21, 37
Offset: 1
Examples
1 -> (1+2)/2 = 3/2 -> (1+5)/2 = 3, so a(1) = 3. 51 -> 49/3 -> 49/3 -> ..., so a(51) = 0.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- P. Schogt, The Wild Number Problem: math or fiction?, arXiv preprint arXiv:1211.6583 [math.HO], 2012. - From _N. J. A. Sloane_, Jan 03 2013
Programs
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Haskell
import Data.Ratio ((%), numerator, denominator) a058971 n = f [n % 1] where f xs@(x:_) | denominator y == 1 = numerator y | y `elem` xs = 0 | otherwise = f (y : xs) where y = (a000203 x') % (a000005 x') x' = numerator x + denominator x -- Reinhard Zumkeller, Aug 02 2012 -
Maple
with(numtheory); f := proc(n) if whattype(n) = integer then sigma(n+1)/sigma[0](n+1) else sigma(numer(n)+denom(n))/sigma[0](numer(n)+denom(n)); fi; end;
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Mathematica
f[x_] := With[{p = Numerator[x], q = Denominator[x]}, DivisorSigma[1, p+q]/DivisorSigma[0, p+q]]; a[n_] := If[ IntegerQ[ r = FixedPoint[f, n, SameTest -> (#1 == #2 || IntegerQ[#2] &)]], r, 0]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Jul 18 2012 *)
Extensions
More terms from Matthew Conroy, Apr 18 2001, who remarks that a(51) = a(655) = a(1039) = 0 are all the zeros of a(n) for n < 10^5
No more zero terms <= 10^6 found by Reinhard Zumkeller, Aug 02 2012
Comments