A057588 Kummer numbers: -1 + product of first n consecutive primes.

1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, 7420738134809, 304250263527209, 13082761331670029, 614889782588491409, 32589158477190044729, 1922760350154212639069

Offset

1,2

Comments

a(n) is congruent to -1 modulo the first n primes. - Michael Engling, Mar 31 2011

Named after the German mathematician Ernst Eduard Kummer (1810-1893). - Amiram Eldar, Jun 19 2021

Links

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

E. E. Kummer, Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist, Monatsber. Preuss. Akad. Wiss., Berlin 1878/9, pp. 777-778. [Collected Papers, II, pp. 669-670, Springer, Berlin-Heidelberg, 1975.] Cited in Mestrovic.

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. - From N. J. A. Sloane, Jun 13 2012

Hisanori Mishima, Factorizations of many number sequences.

Robert G. Wilson v, Explicit factorizations.

Index entries for sequences related to primorial base

Formula

a(n) = A002110(n) - 1. - Altug Alkan, Oct 02 2015

Maple

seq(mul(ithprime(k), k=1..n) - 1, n=1..100); # Muniru A Asiru, Jan 19 2018

Mathematica

Table[Product[Prime[k], {k, 1, n}] - 1, {n, 1, 18}] (* Artur Jasinski, Jan 01 2007 *)
FoldList[Times, 1, Prime[Range[20]]]-1  (* Harvey P. Dale, Apr 17 2011 *)
Table[ChineseRemainder[PadRight[{}, n, -1], Prime[Range[n]]], {n, 20}] (* Harvey P. Dale, Jul 01 2017 *)

Prog

(Haskell)
a057588 = (subtract 1) . product . (flip take a000040_list)
-- Reinhard Zumkeller, Mar 27 2013
(PARI) a(n) = prod(k=1, n, prime(k)) - 1; \\ Michel Marcus, Oct 02 2015
(Python)
from sympy import primorial
def A057588(n): return primorial(n)-1 # Chai Wah Wu, Feb 25 2023

Crossrefs

Cf. A002110, A006862, A057705 (primes), A000040.

Sequence in context: A301878 A331798 A103213 * A030522 A091124 A121143

Adjacent sequences: A057585 A057586 A057587 * A057589 A057590 A057591

Keyword

nonn,easy,nice

Author

Mario Velucchi (mathchess(AT)velucchi.it), Oct 05 2000

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

Status

approved