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A019294
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Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.
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13
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1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15
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OFFSET
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1,2
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COMMENTS
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Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Riele links under A019276.
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
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LINKS
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FORMULA
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Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002
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EXAMPLE
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If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - Don Reble, Jun 23 2005
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MAPLE
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local a, nitr ;
a := 1 ;
nitr := numtheory[sigma](n);
while modp(nitr, n) <> 0 do
nitr := numtheory[sigma](nitr) ;
a := a+1 ;
end do:
return a;
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MATHEMATICA
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f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1, #]&, DivisorSigma[1, n], !Divisible[ #, n]&]], {n, 90}] (* Harvey P. Dale, Mar 04 2015 *)
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PROG
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(PARI) a(n)=if(n<0, 0, c=1; s=n; while(sigma(s)%n>0, s=sigma(s); c++); c)
(PARI) apply( A019294(n, s=n)=for(k=1, oo, (s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020
(Haskell)
a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
(\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
(Magma) a:=[]; f:=func<n|DivisorSigma(1, n)>; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a, s); end for; a; // Marius A. Burtea, Jan 11 2020
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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