A001348 Mersenne numbers: 2^p - 1, where p is prime. (Formerly M2694 N1079)

3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911, 2147483647, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 2305843009213693951, 147573952589676412927, 2361183241434822606847

Offset

1,1

Comments

Mersenne numbers A000225 whose indices are primes. - Omar E. Pol, Aug 31 2008

All terms are of the form 4k-1. - Paul Muljadi, Jan 31 2011

Smallest number with Hamming weight A000120 = prime(n). - M. F. Hasler, Oct 16 2018

The 5th, 8th, 9th, ... terms are not prime. See A000668 for the primes in this sequence. - M. F. Hasler, Nov 14 2018

Except for the first term 3: all prime factors of 2^p-1 must be 1 or -1 (mod 8), and 1 (mod 2p). - William Hu, Mar 10 2024

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 16.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..100

Raymond Clare Archibald, Mersenne's Numbers, Scripta Mathematica, Vol. 3 (1935), pp. 112-119.

John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]

C. K. Caldwell, Mersenne Primes

Will Edgington, Mersenne Page> [from Internet Archive Wayback Machine].

Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 417-431.

Paul Garrett, Lucas-Lehmer criterion for primality of Mersenne numbers, 2010.

Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.

Gabriel Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.

Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.

Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.

Thesaurus.maths.org, Mersenne Number.

Gérard Villemin's Almanach of Numbers, Nombre de Mersenne.

Eric Wegrzynowski, Nombres de Mersenne. [from Internet Archive Wayback Machine]

K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., Vol. 3 (1892), pp. 265-284.

Formula

a(n) = 2^A000040(n) - 1, n >= 1. - Wolfdieter Lang, Oct 26 2014

a(n) = A000225(A000040(n)). - Omar E. Pol, Aug 31 2008

A000668(n) = a(A016027(n)). - Omar E. Pol, Jun 29 2012

Sum_{n>=1} 1/a(n) = A262153. - Amiram Eldar, Nov 20 2020

Product_{n>=1} (1 - 1/a(n)) = A184085. - Amiram Eldar, Nov 22 2022

Maple

A001348 := n -> 2^(ithprime(n))-1: seq (A001348(n), n=1..18);

Mathematica

Table[2^Prime[n]-1, {n, 20}] (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

Prog

(PARI) a(n)=1<<prime(n)-1 \\ Charles R Greathouse IV, Jun 10 2011
(Magma) [2^NthPrime(n)-1: n in [1..30]]; // Vincenzo Librandi, Feb 04 2016
(Python)
from sympy import prime
def a(n): return 2**prime(n)-1
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Mar 28 2022

Crossrefs

Cf. A000043, A000668, A046051, A057951-A057958, A100105, A184085, A262153.

Cf. A000040, A000225. - Omar E. Pol, Aug 31 2008

Cf. A002145, A045326.

Sequence in context: A138864 A105768 A084924 * A006515 A093535 A081093

Adjacent sequences: A001345 A001346 A001347 * A001349 A001350 A001351

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved