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A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.

%I A016047 #64 Mar 12 2025 08:16:29
%S A016047 3,7,31,127,23,8191,131071,524287,47,233,2147483647,223,13367,431,
%T A016047 2351,6361,179951,2305843009213693951,193707721,228479,439,2687,167,
%U A016047 618970019642690137449562111,11447,7432339208719,2550183799,162259276829213363391578010288127
%N A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.
%C A016047 "Mersenne numbers", here, means A001348. Compare to sequence A049479, where "Mersenne numbers" is used in the sense of A000225. - _Lekraj Beedassy_, Jun 11 2009
%C A016047 Submitted new b-file withdrawing last three terms previously submitted.  I had, before submitting that b-file, checked that the smallest known factors of incompletely factored Mersenne numbers was less than the known trial factoring limits recorded by Will Edgington in his LowM.txt file which can be found on his Mersenne page, (see link above.)  I have now discovered that I inadvertently omitted the purported a(203) from that check. - _Daran Gill_, Apr 05 2013
%C A016047 The would-be a(203) corresponds to 2^1237-1 which is currently P70*C303. Trial factoring has only been done to 60 bits, and since its difficulty doubles whenever the bit length is incremented by one, it cannot exhaustively search the space smaller than the sole known 70-digit (231-bit) factor. Probabilistic ECM testing indicates only that it is extremely unlikely that there is any undiscovered factor with digit-size smaller than the high fifties. See GIMPS links. - _Gord Palameta_, Aug 16 2018
%H A016047 Daran Gill, <a href="/A016047/b016047.txt">Table of n, a(n) for n = 1..202</a> (first 95 terms from T. D. Noe)
%H A016047 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes</a>
%H A016047 Will Edgington, <a href="https://web.archive.org/web/20150116003521/http://www.garlic.com/~wedgingt/mersenne.html">Mersenne Page</a>
%H A016047 GIMPS, <a href="https://www.mersenne.org/report_factors/?exp_lo=2&amp;exp_hi=2000&amp;exp_date=2000-01-01&amp;fac_len=&amp;dispdate=1&amp;B1=Get+Factors">PrimeNet Known Factors of Mersenne Numbers (discovered recently for p < 2000)</a>, <a href="https://www.mersenne.org/report_exponent/?exp_lo=1237&amp;full=1">Exponent status of 2^1237 - 1</a>, <a href="https://www.mersenne.org/report_ecm/?txt=0&amp;ecmnof_lo=1237&amp;ecmnof_hi=1237&amp;ecm_lo=1237&amp;ecm_hi=1237">ECM testing status of 2^1237 - 1</a>
%H A016047 Brady Haran and Matt Parker, <a href="https://www.youtube.com/watch?v=lEvXcTYqtKU">How they found the World's Biggest Prime Number - Numberphile</a>, Numberphile video (2016).
%F A016047 a(n) = A020639(A001348(n)). - _Alois P. Heinz_, Oct 01 2024
%p A016047 a:= n-> min(numtheory[factorset](2^ithprime(n)-1)):
%p A016047 seq(a(n), n=1..28);  # _Alois P. Heinz_, Oct 01 2024
%t A016047 a = {}; Do[If[PrimeQ[n], w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b]], {n, 2, 100}]; a (* _Artur Jasinski_, Dec 11 2007 *)
%o A016047 (PARI) forprime(p=2,150,print1(factor(2^p-1)[1,1],", "))
%Y A016047 Cf. A000668 (a subsequence), A003260, A001348, A020639, A046800.
%K A016047 nonn,hard
%O A016047 1,1
%A A016047 _Robert G. Wilson v_