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, 117288381359406970983269, 7858321551080267055879089

Offset: 1

Author

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

Keywords

• nonn
• easy
• nice

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.
• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110, A006862, A057705 (primes), A000040, A000720.

Subsequence of A048103.

Programs

Haskell

a057588 = (subtract 1) . product . (flip take a000040_list)
-- Reinhard Zumkeller, Mar 27 2013

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 *)

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

Formula

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

a(n) = A006862(n) - 2. - Antti Karttunen, Nov 17 2024

Extensions

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