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
Subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i), but neither does p^p divide a(i) when i < A000720(p), as p^p > -1 + A034386(p). - Antti Karttunen, Nov 17 2024
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.
FORMULA
a(n) = A002110(n) - 1. - Altug Alkan, Oct 02 2015
a(n) = A006862(n) - 2. - Antti Karttunen, Nov 17 2024
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
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