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

3, 7, 31, 127, 23, 8191, 131071, 524287, 47, 233, 2147483647, 223, 13367, 431, 2351, 6361, 179951, 2305843009213693951, 193707721, 228479, 439, 2687, 167, 618970019642690137449562111, 11447, 7432339208719, 2550183799, 162259276829213363391578010288127

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• hard

Comments

"Mersenne numbers", here, means A001348. Compare to sequence A049479, where "Mersenne numbers" is used in the sense of A000225. - Lekraj Beedassy, Jun 11 2009

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

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

Links

• Daran Gill, Table of n, a(n) for n = 1..202 (first 95 terms from T. D. Noe)
• C. K. Caldwell, Mersenne Primes
• Will Edgington, Mersenne Page
• GIMPS, PrimeNet Known Factors of Mersenne Numbers (discovered recently for p < 2000), Exponent status of 2^1237 - 1, ECM testing status of 2^1237 - 1
• Brady Haran and Matt Parker, How they found the World's Biggest Prime Number - Numberphile, Numberphile video (2016).

Crossrefs

Cf. A000668 (a subsequence), A003260, A001348, A020639, A046800.

Programs

Maple

a:= n-> min(numtheory[factorset](2^ithprime(n)-1)):
seq(a(n), n=1..28);  # Alois P. Heinz, Oct 01 2024

Mathematica

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 *)

PARI

forprime(p=2,150,print1(factor(2^p-1)[1,1],", "))

Formula

a(n) = A020639(A001348(n)). - Alois P. Heinz, Oct 01 2024