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A036236
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Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.
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67
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1, 0, 3, 4700063497, 6, 19147, 10669, 25, 9, 2228071, 18, 262279, 3763, 95, 1010, 481, 20, 45, 35, 2873, 2951, 3175999, 42, 555, 50, 95921, 27, 174934013, 36, 777, 49, 140039, 56, 2463240427, 110, 477, 697, 91, 578, 623, 156, 2453, 540923, 55, 70, 345119, 287
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OFFSET
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0,3
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COMMENTS
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a(1) = 0, that is, no n exists with 2^n mod n = 1. Proof. Assume that there exists such n > 1. Consider its smallest prime divisor p. Then 2^n == 1 (mod p) implying that the multiplicative order ord_p(2) divides n. However, since ord_p(2) < p and p is the smallest divisor of n, we have ord_p(2) = 1, that is, p divides 2^1 - 1 = 1 which is impossible. [From Max Alekseyev]
Labos Elemer (labos(AT)ana.sote.hu) asked on Sept 27, 2001 if all numbers > 1 eventually appear in A015910, that is, if a(n) > 0 for n > 1.
a(n) > 10^11 for n = 69, 185, 231, 273, 309, 311, 405, 465, 581, 619, 649, 669, 675, 741, 771, 799, 849, 871, 881, 885, 939, 981, ... - Hans Havermann, Apr 19 2007
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REFERENCES
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P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathematique, 28, 1980.
R. K. Guy, Unsolved Problems in Number Theory, Section F10.
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LINKS
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David W. Wilson, Table of n, a(n) for n = 0..1026 (from the Havermann file)
Joe K. Crump, 2^n mod n
Hans Havermann, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) is unknown
Eric Weisstein's World of Mathematics, 2
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FORMULA
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It's obvious that for each k, a(k) > k and we can easily prove that 2^(3^n) = 3^n-1 (mod 3^n). So 3^n is the least k with 2^k mod k = 3^n-1. Hence for each n, a(3^n-1) = 3^n. - Farideh Firoozbakht, Nov 14 2006
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EXAMPLE
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n = 0: 2^1 mod 1 = 0, a(0) = 1;
n = 1: 2^k mod k = 1, no such k exists, so a(1) = 0;
n = 2: 2^3 mod 3 = 2, a(2) = 3;
n = 3: 2^4700063497 mod 4700063497 = 3, a(3) = 4700063497.
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MATHEMATICA
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a = Table[0, {75} ]; Do[ b = PowerMod[2, n, n]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 1, 5*10^9} ]; a
t = Table[0, {1000} ]; k = 1; While[ k < 6500000000, b = PowerMod[2, k, k]; If[b < 1001 && t[[b]] == 0, t[[b]] = k]; k++ ]; t
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PROG
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(PARI) a(n)=if(n==1, return(0)); my(k=n); while(lift(Mod(2, k)^k)!=n, k++); k \\ Charles R Greathouse IV, Oct 12 2011
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CROSSREFS
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Cf. A015910, A015948, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.
Bisections: A122182, A124977.
Sequence in context: A067481 A058433 A154998 * A058447 A216148 A058453
Adjacent sequences: A036233 A036234 A036235 * A036237 A036238 A036239
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KEYWORD
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nonn,nice,changed
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AUTHOR
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David W. Wilson
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EXTENSIONS
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a(3) was first computed by the Lehmers.
More terms from Joe K. Crump (joecr(AT)carolina.rr.com), Sep 04 2000
a(69) = 887817490061261 = 29 * 37 * 12967 * 63809371. [From Hagen von Eitzen, Jul 26 2009]
Edited by Max Alekseyev, Jul 29 2011
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STATUS
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approved
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