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Search: id:A036236
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| A036236 |
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a(n) = least positive number k such that the remainder when 2^k is divided by k is n (the inverse of A015910), or 0 if no such k exists. |
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+0
55
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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No n exists with 2^n mod n = 1.
a(3) was first computed by the Lehmers.
Labos Elemer (labos(AT)ana.sote.hu) asked on Sept 27, 2001 if all numbers > 1 eventually appear in A015910.
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 (pxp(AT)rogers.com), Apr 19 2007
a(69) = 887817490061261 = 29 * 37 * 12967 * 63809371. [From Hagen von Eitzen (math(AT)von-eitzen.de), Jul 26 2009]
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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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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) > 10^11
Topology Q+A Board, Prove that 2^n = 1 (mod n) is impossible for an integer n > 1
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
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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 (mymontain(AT)yahoo.com), Nov 14 2006
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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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CROSSREFS
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Cf. A015910, A015948.
Cf. A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.
Bisections: A122182, A124977.
Sequence in context: A067481 A058433 A154998 this_sequence A058447 A058453 A058471
Adjacent sequences: A036233 A036234 A036235 this_sequence A036237 A036238 A036239
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KEYWORD
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nonn,nice
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
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More terms from Joe K. Crump (joecr(AT)carolina.rr.com), Sep 04 2000
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