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 A036236 Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists. 68
 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; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. - Max Alekseyev Labos Elemer asked on Sep 27 2001 if all numbers > 1 eventually appear in A015910, that is, if a(n) > 0 for n > 1. Obviously, k > n. - Daniel Forgues, Jul 06 2015 REFERENCES P. Erdős 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. LINKS David W. Wilson, Table of n, a(n) for n = 0..1026 (from the Havermann file) Joe K. Crump, 2^n mod n Eric Weisstein's World of Mathematics, 2 FORMULA 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 EXAMPLE 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. MATHEMATICA 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 (* Second program: *) 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 nk[n_] := Module[ {k}, k = 1; While[PowerMod[2, k, k] != n, k++]; k] Join[{1, 0}, Table[nk[i], {i, 2, 46}]]  (* Robert Price, Oct 11 2018 *) PROG (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 CROSSREFS Cf. A015910, A015948, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821. Bisections: A122182, A124977. Sequence in context: A067481 A058433 A154998 * A235357 A260002 A058447 Adjacent sequences:  A036233 A036234 A036235 * A036237 A036238 A036239 KEYWORD nonn,nice AUTHOR EXTENSIONS 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. - Hagen von Eitzen, Jul 26 2009 Edited by Max Alekseyev, Jul 29 2011 STATUS approved

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Last modified June 12 19:06 EDT 2021. Contains 344959 sequences. (Running on oeis4.)