A001348 Mersenne numbers: 2^p - 1, where p is prime.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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, 9th, 10th, ... terms are not prime. See A000668 and A065341 for the primes and for the composites in this sequence. - M. F. Hasler, Nov 14 2018 [corrected by Jerzy R Borysowicz, Apr 08 2025]

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.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
• 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.
• Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 1, p. 531.
• 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.

Crossrefs

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

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

Cf. A002145, A045326.

Programs

Magma

[2^NthPrime(n)-1: n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

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 *)

PARI

a(n)=1<Charles R Greathouse IV, Jun 10 2011

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

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