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Search: id:A019294
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| A019294 |
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Number (>0) of iteration of sigma (A000203) required to reach a value divisible by n when starting with n. |
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7
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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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Let sigma_m (n) be result of applying sum-of-divisors function m times to n; let m(n) = min m such that n divides sigma_m (n); let k(n) = sigma_{m(n)}(n)/n; sequence gives m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Reile links under A019276.
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REFERENCES
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Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..400
Experimental Mathematics, Home Page
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FORMULA
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Conjecture : lim n -> infinity 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 101-st 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. - Dan Reble, Jun 23 2005
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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}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 24 2005)
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PROGRAM
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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)
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CROSSREFS
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Cf. A019295, A000396, records in A019276.
Sequence in context: A136620 A139548 A108445 this_sequence A057037 A076920 A020774
Adjacent sequences: A019291 A019292 A019293 this_sequence A019295 A019296 A019297
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Labos E. (labos(AT)ana.sote.hu), Jun 20 2001
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