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%I #54 Jul 07 2023 16:44:37
%S 1,3,7,5,31,1,127,17,73,11,23,13,8191,43,151,257,131071,19,524287,41,
%T 337,683,47,241,601,2731,262657,29,233,331,2147483647,65537,599479,
%U 43691,71,37,223,174763,79,61681,13367,5419,431,397,631,2796203,2351,97,4432676798593,251,103,53,6361,87211
%N 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 If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);
%C a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.
%C If a(n) > 1 then a(n) is the index where n occurs first in A014664. - _M. F. Hasler_, Feb 21 2016
%C 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 Robert G. Wilson v, <a href="/A112927/b112927.txt">Table of n, a(n) for n = 1..1206</a>
%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem">Zsigmondy's theorem</a>
%o (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 Cf. A002326, A064078, A001122, A115591.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Aug 25 2008
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