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A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

%I A112927 #57 Dec 28 2024 16:18:54
%S A112927 1,3,7,5,31,1,127,17,73,11,23,13,8191,43,151,257,131071,19,524287,41,
%T A112927 337,683,47,241,601,2731,262657,29,233,331,2147483647,65537,599479,
%U A112927 43691,71,37,223,174763,79,61681,13367,5419,431,397,631,2796203,2351,97,4432676798593,251,103,53,6361,87211
%N A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.
%C A112927 If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);
%C A112927 a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.
%C A112927 If a(n) > 1 then a(n) is the index where n occurs first in A014664. - _M. F. Hasler_, Feb 21 2016
%C A112927 Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - _Jeppe Stig Nielsen_, Aug 31 2020
%H A112927 Robert G. Wilson v, <a href="/A112927/b112927.txt">Table of n, a(n) for n = 1..1206</a>
%H A112927 Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%H A112927 Will Edgington, <a href="https://web.archive.org/web/20111107151843/http://www.garlic.com/~wedgingt/factoredM.txt">Factored Mersenne Numbers</a> [from Internet Archive Wayback Machine]
%H A112927 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem">Zsigmondy's theorem</a>
%o A112927 (PARI) A112927(n,f=factor(2^n-1)[,1])=!for(i=1,#f,znorder(Mod(2,f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1,0)[,1]. - _M. F. Hasler_, Feb 21 2016
%Y A112927 Cf. A002326, A064078, A001122, A115591.
%Y A112927 Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).
%K A112927 nonn
%O A112927 1,2
%A A112927 _Vladimir Shevelev_, Aug 25 2008