A001348 -id:A001348 - cp's OEIS frontend

A350702 Primes p such that 14*p + 1 divides 2^p - 1.

929, 1433, 2393, 2609, 2657, 4373, 4793, 6029, 7529, 10133, 10433, 10949, 10973, 13049, 13109, 16829, 18869, 20873, 22349, 23417, 24137, 26717, 27737, 27893, 28433, 28517, 30977, 33809, 33857, 37217, 38189, 38237, 39209, 39749, 41453, 41813, 42569, 43313, 43613

Offset: 1

Karl-Heinz Hofmann, Jan 27 2022

nonn

Known divisors of Mersenne(p) (2^p-1 or Mp for short) are of the form 2*k*p+1. See crossrefs for other k's. If k == 2 (mod 4), there are no such divisors in general. Here k is 14/2 = 7.

			See LINKS for example of a(13).

Karl-Heinz Hofmann, Mersenne(p) table with k = 7 and, if they exist, additional k < 7 plus their corresponding factors.
mersenne.ca, a(13) = M10973 Mersenne number exponent details.
mersenne.ca, a(1..995) at GIMPS with k = 7 and Exponent 3..2000000.

Cf. A002515 (k = 1), A188130 (k = 3), A122095 (k = 4), A188133 (k = 5).

Cf. A000040, A001348.

Select[Range[45000], PrimeQ[#] && PowerMod[2, #, 14*# + 1] == 1 &] (* Amiram Eldar, Jan 27 2022 *)

forprime(p=1, 1e6, if (Mod(2, p*14+1)^p==1, print1(p,", ")))

from sympy import sieve
print([p for p in sieve[1:1000000] if pow(2, p, 14*p+1) == 1])

{p = A000040(i): 14*p+1 | A001348(i)}.

A037577 Base 5 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

Original entry on oeis.org

1, 8, 41, 208, 1041, 5208, 26041, 130208, 651041, 3255208, 16276041, 81380208, 406901041, 2034505208, 10172526041, 50862630208, 254313151041, 1271565755208, 6357828776041, 31789143880208, 158945719401041, 794728597005208

Offset: 1

Clark Kimberling

This is a particular case of the generalized sequence a(n)=((A^n) - B)/(A-B). Sometimes the primes of this form are of interest, see A001348, A014224, A028491. - Ctibor O. Zizka, Apr 15 2008

			a(1) = (5-1)/3 = 1, a(2) = (5^2-1)/3 = 8. - _Philippe Deléham_, Nov 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,1,-5).

CoefficientList[Series[(3 x + 1)/((x - 1) (x + 1) (5 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 21 2013 *)

a(n) = (5^n - 2)/3 for n odd ; a(n) = (5^n - 1)/3 for n even. - Ctibor O. Zizka, Apr 15 2008
a(n) = floor(5^n/3). - Gary Detlefs, Sep 06 2010
a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3). - Charles R Greathouse IV, Jan 15 2011
G.f.: x*(3*x+1) / ((x-1)*(x+1)*(5*x-1)). - Colin Barker, Dec 27 2012

First formula corrected by Philippe Deléham, Nov 14 2013

A053159 Numbers n such that n+cototient(n) is a power of 2.

Original entry on oeis.org

1, 3, 7, 10, 20, 31, 40, 80, 127, 160, 320, 322, 640, 644, 1280, 1288, 2560, 2576, 5120, 5152, 8191, 10240, 10304, 20480, 20608, 40960, 41216, 81920, 82432, 131071, 163840, 164864, 327680, 329728, 333634, 524287, 655360, 659456, 667268, 1310720, 1318912

Offset: 1

Labos Elemer, Feb 29 2000

nonn

See especially A053579 and also A053576, A053577.

			Mersenne primes are a proper subset of this sequence: A(M)=2M-M+1=M+1=2^p

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000043, A000668, A001348.

print(1); for(n=3, 10^9, if(omega(2*n-eulerphi(n))==1, print(n))) /* Donovan Johnson, Apr 04 2013 */

a(n)+A051953(n) = 2*a(n)-A000010(n) = 2^w for some w.

More terms from Reiner Martin, Dec 24 2001

A056743 a(n) = phi(2^prime(n) - 1)/prime(n); a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 6, 18, 176, 630, 7710, 27594, 356960, 18407808, 69273666, 3697909056, 53630700752, 204064589160, 2992477516800, 169917983040000, 9770466930024800, 37800705069076950, 2202596295934991760

Offset: 0

Robert G. Wilson v, Aug 14 2000

nonn

Cf. A000020, A064535.

with numtheory; A056743 := proc(n) phi( 2^ithprime(n) - 1 )/ithprime(n); end;

Phi( A001348) / A000040. Table[EulerPhi[(2^Prime[n] - 1)]/Prime[n], {n, 1, 25}]

A089159 If Mersenne numbers have 3 or more factors, then list the third factor.

Original entry on oeis.org

2089, 2099863, 13264529, 20394401, 212885833, 9361973132609, 1113491139767, 65993, 165799, 1654058017289, 110211473, 70084436712553223, 1489459109360039866456940197095433721664951999121, 7648337, 39940132241, 14732265321145317331353282383

Offset: 1

Cino Hilliard, Dec 06 2003, corrected Nov 16 2006

nonn

			The 10th Mersenne number 2^29 - 1 = 233*1103*2089 and 2089 is the third prime factor. Notice these factors are congruent to 1 (mod 29).

Amiram Eldar, Table of n, a(n) for n = 1..144
Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

Cf. A000043, A001348, A016047, A089158, A135977, A344515.

mersenne2(n) = { c=0; forprime(x=2, n, c++; y = 2^x-1; f=ifactor(y); if(length(f)>=3, print1(f[3]","); ) ) }
ifactor(n) = { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

A Mersenne number (A001348) is a number of the form 2^p - 1 where p is prime.

a(16) from Amiram Eldar, Jul 11 2024

A103902 Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.

Original entry on oeis.org

8191, 131071, 524287, 2147483647

Offset: 1

Jonathan Sondow, Feb 20 2005

Only four terms are known.

The first four Mersenne primes (p=2^q-1 in A000668) are double Mersenne primes, i.e., in A103901. The next four yield a composite M(p) and therefore are in this sequence. The next larger Mersenne prime p = A000668(9) has already 19 digits and is much too large to enable us, as of today, to test the primality of 2^p-1 (which would require over 10^8 gigabytes just to be stored in binary). This explains that only 4 terms are known of this sequence and of A103901; for all the 30+ remaining members of A000668 it is not known whether they belong to A103901 or to this sequence A103902. - M. F. Hasler, Jan 21 2015

			M(13) = 8191 is a Mersenne prime and M(1891) is composite, so 1891 is a member.

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.

C. K. Caldwell, Mersenne Primes: Conjectures and Unsolved Problems
Eric Weisstein's World of Mathematics, Double Mersenne Number
Wikipedia, Mersenne prime

Cf. A000043, A000668, A001348, A077585, A077586, A103901.

is(n)=isprime(2^n-1) && !isprime(2^(2^n-1)-1) \\ Charles R Greathouse IV, Jan 24 2015

A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

Original entry on oeis.org

2, 5, 10, 17, 30, 47, 66, 97, 158, 247, 354, 481, 1002, 1609, 2888, 5091, 7372, 10589, 14842, 19265, 28954, 38895, 50108, 70045, 91746, 114955, 159452, 245695, 356198, 488247, 704338, 1461177, 2320610, 3578397, 4976666, 7952887, 10974264, 17946857, 31413774, 52409785, 76446368, 102411319, 132813776, 165396433, 202553100, 245196901, 288309510

Offset: 1

Jonathan Vos Post, Aug 28 2005

nonn

Prime cumulative sum of primes p such that 2^p - 1 is a Mersenne prime include: a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 47, a(8) = 97, a(14) = 1609, a(18) = 10589. After 1, all such indices x of prime a(x) must be even.

			a(1) = 2, since 2^2-1 = 3 is a Mersenne prime.
a(2) = 2 + 3 = 5, since 2^3-1 = 7 is a Mersenne prime.
a(3) = 2 + 3 + 5 = 10, since 2^5-1 = 31 is a Mersenne prime.
a(4) = 2 + 3 + 5 + 7 = 17, since 2^7-1 = 127 is a Mersenne prime; 17 itself is prime (in fact a p such that 2^p-1 is a Mersenne prime).
a(18) = 2 + 3 + 5 + 7 + 13 + 17 + 19 + 31 + 61 + 89 + 107 + 127 + 521 + 607 + 1279 + 2203 + 2281 + 3217 = 10589 (which is prime).

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Gord Palameta)

Cf. A000043, A000668 for the Mersenne primes, A001348, A046051, A057951-A057958.

Accumulate[Select[Range[3000], PrimeQ[2^# - 1] &]] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Accumulate@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)

a(n) = Sum_{i=1..n} A000043(i).

a(38)-a(47) from Gord Palameta, Jul 21 2018

A125954 Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 11, 2, 5, 11, 2, 2, 5, 71, 2, 3, 2, 2, 167, 2, 17, 3, 2, 197, 149, 2, 2, 3, 3, 2, 2267, 2, 2, 3, 3, 2, 29, 2, 2531, 167, 2, 7, 3, 3, 2, 61, 2, 2, 11, 2, 2, 157, 2, 5, 7, 7, 149, 3, 5, 2, 379, 2, 41, 3, 2, 2, 3, 79, 11, 3, 2, 2, 97, 3, 2, 3, 3, 2, 1321, 2, 17, 31, 2, 61

Offset: 0

Alexander Adamchuk, Feb 07 2007

All terms are primes.
a(n) = 2 for n = {1,2,4,5,7,8,10,13,14,17,19,20,22,...} = A067076 Numbers n such that 2n+3 is a prime.
a(34),...,a(40) = {2,2,3,3,2,29,2}.
a(42),...,a(80) = {167,2,7,3,3,2,61,2,2,11,2,2,157,2,5,7,7,149,3,5,2,379,2,41,3,2,2,3,79,11,3,2,2,97,3,2,3,3,2}.
a(82),...,a(90) = {2,17,31,2,61,7,2,2,5}.
a(93),...,a(95) = {383,2,2}.
a(97),...,a(100) = {2,2,5,7}.
a(102),...,a(124) = {13,11,2,5,5,17,3,103,2,19,2,2,3,2,31,37,2,2,3,3,7,3,2}.
a(127),...,a(131) = {2,61,31,2,157}.
a(133),...,a(142) = {2,2,7,3,2,13,2,2,7,3}.
a(144),...,a(146) = {173,2,11}.
a(148),...,a(150) = {3,17,107}.
a(n) is currently unknown for n = {33,41,81,91,92,96,101,125,126,132,143,147,...}.

Cf. A067076.
Cf. A000043 = Primes p such that 2^p - 1 is prime.
Cf. A001348 = Mersenne numbers: 2^p - 1, where p is prime.
Cf. A057468 = numbers n such that 3^n - 2^n is prime.
Cf. A125958 = Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

Do[k = 1; While[ !PrimeQ[((2n+1)^k - 2^k)/(2n-1)], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Mar 29 2007 *)
lnk[n_]:=Module[{k=1},While[!PrimeQ[((2n+1)^k-2^k)/(2n-1)],k++];k]; Array[ lnk,90] (* Harvey P. Dale, May 19 2012 *)

More terms from Ryan Propper, Mar 29 2007

A128311 Remainder upon division of 2^(n-1)-1 by n.

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 7, 3, 1, 0, 7, 0, 1, 3, 15, 0, 13, 0, 7, 3, 1, 0, 7, 15, 1, 12, 7, 0, 1, 0, 31, 3, 1, 8, 31, 0, 1, 3, 7, 0, 31, 0, 7, 30, 1, 0, 31, 14, 11, 3, 7, 0, 13, 48, 15, 3, 1, 0, 7, 0, 1, 3, 63, 15, 31, 0, 7, 3, 21, 0, 31, 0, 1, 33, 7, 8, 31, 0, 47, 39, 1, 0, 31, 15, 1, 3, 39, 0, 31, 63

Offset: 1

M. F. Hasler, May 04 2007

nonn

By Fermat's little theorem, if p > 2 is prime, then 2^(p-1) == 1 (mod p), thus a(p)=0. If a(n)=0, then n may be only pseudoprime, as for n = 341 = 11*31 [F. Sarrus, 1820].

See A001567 for the list of all pseudoprimes to base 2, i.e., composite numbers which have a(n) = 0, also called Sarrus or Poulet numbers. Carmichael numbers A002997 are pseudoprimes to all (coprime) bases b >= 2. - M. F. Hasler, Mar 13 2020

			a(1)=0 since any integer == 0 (mod 1);
a(2)=1 since 2^1-1 == 1 (mod 2),
a(3)=0 since 3 is a prime > 2,
a(4)=3 since 2^3-1 = 7 == 3 (mod 4);
a(341)=0 since 341=11*31 is a Sarrus number.

T. D. Noe, Table of n, a(n) for n = 1..2048
Wikipedia, Fermat's little theorem

Cf. A001348 (Mersenne numbers), A001567 (Sarrus numbers: pseudoprimes to base 2), A002997 (Carmichael numbers), A084653, A001220 (Wieferich primes).

Table[Mod[2^(n-1)-1,n],{n,100}] (* Harvey P. Dale, Dec 22 2012 *)

PARI
```
a(n)=(1<<(n-1)-1)%n
```

apply( {A128311(n)=lift(Mod(2,n)^(n-1)-1)}, [1..99]) \\ Much more efficient when n becomes very large. - M. F. Hasler, Mar 13 2020

def A128311(n): return (pow(2,n-1,n)-1)%n # Chai Wah Wu, Jul 06 2022

a(n) = M(n-1) - n floor( M(n-1)/n ) = M(n-1) - max{ k in nZ | k <= M(n-1) } where M(k)=2^k-1.

A184085 Decimal expansion of product_{p=primes} (1-1/(2^p-1)).

Original entry on oeis.org

5, 4, 8, 3, 0, 0, 8, 3, 1, 2, 8, 2, 0, 9, 8, 4, 0, 7, 6, 7, 7, 6, 4, 0, 4, 9, 1, 5, 2, 2, 6, 7, 3, 1, 5, 4, 4, 9, 7, 0, 1, 9, 9, 4, 6, 1, 0, 0, 1, 8, 5, 0, 9, 9, 4, 5, 1, 3, 7, 3, 8, 5, 9, 2, 8, 5, 7, 9, 7, 7, 4, 2, 6, 6, 3, 7, 4, 1, 6, 2, 1, 7, 2, 4, 5, 4, 9, 6, 4, 3, 0, 1, 2, 2, 2, 6, 9, 3, 1, 9, 6, 3, 2

Offset: 0

R. J. Mathar, Jan 09 2011

			(1-1/3) *(1-1/7) *(1-1/31) *(1-1/127) *(1-1/2047) * ... = 0.5483008312820984076776404...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 17.

digits = 103; m0 = 10; dm = 10; f[m_] := f[m] = Product[p = Prime[n]; 1 - 1/(2^p - 1), {n, 1, m}]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits+2] != RealDigits[f[m - dm], 10, digits+2], m = m + dm]; RealDigits[f[m], 10, digits] // First (* Jean-François Alcover, Oct 14 2014 *)

Equals product_{p in A000040} (1-1/(2^p-1)) = product_{n>=1} (1-1/A001348(n)).

More digits from Jean-François Alcover, Oct 14 2014

cp's OEIS Frontend

A350702 Primes p such that 14*p + 1 divides 2^p - 1.

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A037577 Base 5 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

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A053159 Numbers n such that n+cototient(n) is a power of 2.

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A056743 a(n) = phi(2^prime(n) - 1)/prime(n); a(0) = 0 by convention.

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Maple

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A089159 If Mersenne numbers have 3 or more factors, then list the third factor.

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A103902 Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.

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A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

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A125954 Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.