A049479 -id:A049479 - cp's OEIS frontend

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

Robert G. Wilson v

"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

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).

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

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

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 *)

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

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

A053285 Totient of 2^n+1.

Original entry on oeis.org

1, 2, 4, 6, 16, 20, 48, 84, 256, 324, 800, 1364, 3840, 5460, 12544, 19800, 65536, 87380, 186624, 349524, 986880, 1365336, 3345408, 5592404, 16515072, 20250000, 52306176, 84768120, 252645120, 351847488, 760320000, 1431655764, 4288266240, 5632621632, 13628740608

Offset: 0

Labos Elemer, Mar 03 2000

nonn

			It is a power of 2 iff n is a Fermat prime.

Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..300 from Robert Israel; terms 301..1062 from Amiram Eldar)

Cf. A000010, A000225, A002587, A000051, A046798, A046799, A049479, A051953, A054992.

[EulerPhi(2^n+1) : n in [1..40]]; // Vincenzo Librandi, Aug 12 2015

seq(numtheory:-phi(2^n+1), n=0..50); # Robert Israel, Aug 12 2015

Table[EulerPhi[2^n + 1], {n, 35}] (* Vincenzo Librandi, Aug 12 2015 *)

vector(40, n, eulerphi(2^n+1)) \\ Michel Marcus, Aug 12 2015

a(n) = A000010(A000051(n)).

a(0)=1 prepended by Alois P. Heinz, Aug 12 2015

A361438 Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

Original entry on oeis.org

1, 1, 3, 1, 7, 1, 3, 5, 15, 1, 31, 1, 3, 7, 9, 21, 63, 1, 127, 1, 3, 5, 15, 17, 51, 85, 255, 1, 7, 73, 511, 1, 3, 11, 31, 33, 93, 341, 1023, 1, 23, 89, 2047, 1, 3, 5, 7, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 1, 8191, 1, 3, 43, 127, 129, 381, 5461, 16383

Offset: 1

Seiichi Manyama, Mar 12 2023

			Triangle begins:
  1;
  1,   3;
  1,   7;
  1,   3,  5,   15;
  1,  31;
  1,   3,  7,    9, 21, 63;
  1, 127;
  1,   3,  5,   15, 17, 51,  85,  255;
  1,   7, 73,  511;
  1,   3, 11,   31, 33, 93, 341, 1023;
  1,  23, 89, 2047;

Seiichi Manyama, Rows n = 1..64, flattened
Index entries for sequences related to divisors of numbers

Subsequence of A027750.
Cf. A000225, A049479 (2nd column), A075708 (row sums).
Cf. A374237 (analogous for 2^n + 1).

T:= n-> sort([numtheory[divisors](2^n-1)[]])[]:
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 20 2024

Divisors[2^Range[15] - 1] (* Paolo Xausa, Jul 02 2024 *)

A136031 Largest prime factor of composite Mersenne numbers.

Original entry on oeis.org

89, 178481, 2089, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 761838257287, 212885833, 9361973132609, 1113491139767, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A049479, A016047, A003260, A136030.

FactorInteger[2^#-1][[-1,1]]&/@Select[Prime[Range[40]],!PrimeQ[2^#-1]&] (* Harvey P. Dale, May 05 2012 *)
With[{c=Complement[Prime[Range[PrimePi[200]]],MersennePrimeExponent[ Range[ 15]]]},Table[ FactorInteger[ 2^p-1][[-1,1]],{p,c}]] (* Harvey P. Dale, Sep 06 2021 *)

a(n) >= A089158(n). - R. J. Mathar, May 01 2008

Corrected by Harvey P. Dale, May 05 2012

A136030 Smallest prime factor of composite Mersenne numbers.

Original entry on oeis.org

23, 47, 233, 223, 13367, 431, 2351, 6361, 179951, 193707721, 228479, 439, 2687, 167, 11447, 7432339208719, 2550183799, 745988807, 3391, 263, 32032215596496435569, 5625767248687, 86656268566282183151, 18121, 852133201, 150287, 2349023, 730753, 359, 43441, 383

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

Max Alekseyev, Table of n, a(n) for n = 1..188

Cf. A049479, A016047.

a = {}; Do[If[PrimeQ[n] && !PrimeQ[2^n - 1], w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b]], {n, 2, 130}]; a

lista() = {vi = readvec("c:/gp/bfiles/b054723.txt"); vm = vector(#vi, i, 2^vi[i]-1); for (i=1, #vm, print1(factor(vm[i])[1, 1], ", "););} \\ Michel Marcus, May 14 2014

More terms from Michel Marcus, May 14 2014
Terms to a(150) in b-file from Charles R Greathouse IV, May 14 2014
a(151)-a(188) in b-file added at the suggestion of Eric Chen by Max Alekseyev, Apr 25 2022

A186283 Least number k such that k*n+1 is a prime dividing 2^n-1.

Original entry on oeis.org

1, 2, 1, 6, 1, 18, 2, 8, 1, 2, 1, 630, 3, 2, 1, 7710, 1, 27594, 2, 6, 1, 2, 10, 24, 105, 9728, 1, 8, 1, 69273666, 8, 18166, 1285, 2, 1, 6, 4599, 2, 1, 326, 1, 10, 2, 14, 1, 50, 2, 90462791808, 5, 2, 1, 120, 1615, 16, 2, 568, 1, 3050, 1, 37800705069076950, 11545611, 2, 4, 126, 1, 2891160, 2, 145690999102, 1

Offset: 2

Bill McEachen, Feb 16 2011

nonn

The smallest prime factor of 2^n-1 of the form k*n+1 is A186522(n).

By Fermat's little theorem, a(n) = 1 if and only if n+1 is an odd prime. Further, for prime p, a(p) = 2 if and only if p is in A002515. - Thomas Ordowski, Sep 03 2017

			For n=8, 2^n-1 = 255 = 3 * 5 * 17.  The smallest prime factor of the form k*n+1 is 17 = 2*8+1. Hence, a(8) = 2.

Kenneth H. Rosen, Elementary Number Theory and Its Applications, 3rd Ed, Theorem 6.12, p. 225

Max Alekseyev, Table of n, a(n) for n = 2..1236 (terms to a(300) from David A. Corneth)
Will Edgington, Mersenne Page [from Internet Archive Wayback Machine]
Eric W. Weisstein, MathWorld: Mersenne Number
Eric W. Weisstein, MathWorld: Mersenne Prime

Cf. A000225, A049479, A079324.

Table[p=First/@FactorInteger[2^n-1]; (Select[p, Mod[#1,n] == 1 &, 1][[1]] - 1)/n, {n, 2, 70}]

a(n) = {if(isprime(n+1),return(1)); my(f = factor(2^n - 1)[,1]); for(i=1,#f, if(f[i]%n == 1, return((f[i]-1) / n)))} \\ David A. Corneth, Sep 03 2017

a(n) = (A186522(n)-1)/n.

A367004 a(n) is the smallest prime factor of n*2^n+1.

Original entry on oeis.org

3, 3, 5, 5, 7, 5, 3, 3, 11, 7, 13, 13, 3, 3, 17, 17, 5, 11, 3, 3, 23, 13, 5, 5, 3, 3, 7, 29, 31, 17, 3, 3, 47, 19, 37, 37, 3, 3, 41, 41, 13, 23, 3, 3, 11, 5, 7, 7, 3, 3, 53, 7, 5591, 29, 3, 3, 5, 31, 37, 61, 3, 3, 5, 5, 67, 5, 3, 3, 7, 37, 11, 41, 3, 3, 149

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..914 (terms 1..350 from Robert Israel)

Cf. A002064, A005849, A020639, A049479, A367002, A367003, A367005.

seq(min(numtheory:-factorset(n*2^n+1)), n=1..100); # Robert Israel, Nov 09 2023

Table[FactorInteger[n 2^n+1][[1,1]],{n,80}] (* Harvey P. Dale, Aug 14 2024 *)

a(n) = A020639(A002064(n)).

A236769 Numbers n such that lpf(2^n -1) < lpf(2^lpf(n) -1).

Original entry on oeis.org

55, 77, 161, 169, 221, 275, 299, 323, 377, 385, 391, 437, 481, 493, 539, 551, 559, 605, 611, 629, 689, 697, 703, 715, 731, 779, 793, 799, 817, 847, 893, 901, 923, 935, 949, 1001, 1007, 1027, 1045, 1073, 1079, 1121, 1127, 1147, 1159, 1241, 1265, 1271, 1273, 1309

Offset: 1

Thomas Ordowski, Jan 31 2014

nonn

The numbers n for which A049479(n) < A049479(lpf(n)), where lpf(n) = A020639(n). All other n satisfy the equality (in particular all primes).

All terms are odd and composite. - Chai Wah Wu, Oct 04 2019

Chai Wah Wu, Table of n, a(n) for n = 1..111

Cf. A049479 (a question in the third comment).

lpf(n) = vecmin(factor(n)[, 1]);
lista() = {my(vlpfmp = readvec("A049479.log")); for (i=2, #vlpfmp, if (vlpfmp[i] < vlpfmp[lpf(i)], print1(i, ", ")););} \\ Michel Marcus, Jan 31 2014

More terms from Michel Marcus, Jan 31 2014

More terms from Chai Wah Wu, Oct 04 2019

A263686 Smallest prime factor of double Mersenne numbers.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

A292559 Composite numbers m such that lpf(2^m - 1) == 1 (mod m).

Original entry on oeis.org

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1343, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2201, 2413, 2533, 2603, 2759, 2951, 3097, 3131, 3173, 3193, 3281, 3379, 3599, 3721, 3743, 3791, 3937, 3953, 4043, 4223, 4309, 4331, 4369, 4607, 4619, 4811, 4867, 4883, 4981, 5111, 5177, 5263, 5429, 5567, 5699

Offset: 1

Michel Marcus and Thomas Ordowski, Sep 19 2017

All terms are coprime to 2, 3, 5, 7, 11. - Robert Israel, Sep 20 2017
If p = lpf(2^m - 1) and A002326((p-1)/2) = m composite, then m is in this sequence. - Thomas Ordowski, Sep 20 2017
Conjecture: there are no numbers k in this sequence such that, for each prime factor q of 2^k - 1, q == 1 (mod k). - Thomas Ordowski, Sep 20 2017
Note: if all prime factors q of 2^k - 1 are q == 1 (mod k), then 2^k - 1 == 1 (mod k), thus 2^k == 2 (mod k), so k is a pseudoprime. The pseudoprime k = a(42) = 4369 = 17*257 is not a counterexample to this conjecture. A pseudoprime k = P*Q such that both 2^P - 1 and 2^Q - 1 are primes would be a counterexample, but the known Mersenne primes do not give such k. - Thomas Ordowski, Oct 02 2017
If lpf(2^n - 1) == 1 (mod n), then gpf(2^n - 1) == 1 (mod n). Cf. A291855. - Thomas Ordowski, Oct 20 2017
Composites m such that lpf(2^m - 1)*gpf(2^m - 1) is a Fermat pseudoprime to base 2, i.e., is in A214305. - Thomas Ordowski, Oct 29 2017

Subsequence of A236769.

Cf. A002326, A049479, A292834, A298076.

searchMax = 1000; Complement[Select[Range[searchMax], Mod[FactorInteger[2^# - 1][[1, 1]], #] == 1 &], Prime[Range[PrimePi[searchMax]]]] (* Alonso del Arte, Sep 19 2017 *)

lista(nn) = forcomposite(n=1, nn, sp = factor(2^n-1)[1,1]; if ((sp % n) == 1, print1(n, ", "))); \\ Michel Marcus, Sep 19 2017

A049479(m) == 1 (mod m).

a(10)-a(54) from Charles R Greathouse IV, Sep 19 2017

cp's OEIS Frontend

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

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A053285 Totient of 2^n+1.

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A361438 Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

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A136031 Largest prime factor of composite Mersenne numbers.

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A136030 Smallest prime factor of composite Mersenne numbers.

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A186283 Least number k such that k*n+1 is a prime dividing 2^n-1.

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A367004 a(n) is the smallest prime factor of n*2^n+1.

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