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A001917
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(p-1)/x, where p = prime(n) and x = ord(2,p), the smallest positive integer such that 2^x == 1 (mod p).
(Formerly M0069 N0022)
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33
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1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1, 1, 2, 1, 9, 2, 2, 1, 1, 10, 6, 6, 1, 2, 6, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 2
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OFFSET
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2,3
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COMMENTS
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Also number of cycles in permutations constructed from siteswap juggling pattern 1234...p.
Also the number of irreducible polynomial factors for the polynomial (x^p-1)/(x-1) over GF(2), where p is the n-th prime. - V. Raman, Oct 04 2012
The sequence is unbounded: for any value of M, there exists an element of the sequence divisible by M. See the proof by David Speyer below. - Shreevatsa R, May 24 2013
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REFERENCES
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M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 131.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = A006694((p_n-1)/2) where p_n is the n-th odd prime.
Conjecture: k*a(n) = A006694(((p_n)^k-1)/2). (End)
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MAPLE
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with(numtheory); [seq((ithprime(n)-1)/order(2, ithprime(n)), n=2..130)];
with(group); with(numtheory); gen_rss_perm := proc(n) local a, i; a := []; for i from 1 to n do a := [op(a), ((2*i) mod (n+1))]; od; RETURN(a); end; count_of_disjcyc_seq := [seq(nops(convert(gen_rss_perm(ithprime(j)-1), 'disjcyc')), j=2..)];
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MATHEMATICA
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a6694[n_] := Sum[ EulerPhi[d] / MultiplicativeOrder[2, d], {d, Divisors[2n + 1]}] - 1; a[n_] := a6694[(Prime[n]-1)/2]; Table[ a[n], {n, 2, 104}] (* Jean-François Alcover, Dec 14 2011, after Vladimir Shevelev *)
Table[p = Prime[n]; (p - 1)/MultiplicativeOrder[2, p], {n, 2, 100}] (* T. D. Noe, Apr 11 2012 *)
ord[n_]:=Module[{x=1}, While[PowerMod[2, x, n]!=1, x++]; (n-1)/x]; ord/@ Prime[ Range[ 2, 110]] (* Harvey P. Dale, Jun 25 2014 *)
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PROG
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(Magma) [ (p-1)/Modorder(2, p) where p is NthPrime(n): n in [2..100] ]; // Klaus Brockhaus, Dec 09 2008
(PARI) {for(n=2, 100, p=prime(n); print1((p-1)/znorder(Mod(2, p)), ", "))} \\ Klaus Brockhaus, Dec 09 2008
(Python)
from sympy import prime, n_order
p = prime(n)
return 1 if n == 2 else (p-1)//n_order(2, p) # Chai Wah Wu, Jan 15 2020
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CROSSREFS
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Cf. A006694 gives cycle counts of such permutations constructed for all odd numbers.
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KEYWORD
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nonn,easy,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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