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%I #61 Feb 26 2023 02:22:21
%S 1,5,29,209,2309,30029,510509,9699689,223092869,6469693229,
%T 200560490129,7420738134809,304250263527209,13082761331670029,
%U 614889782588491409,32589158477190044729,1922760350154212639069
%N Kummer numbers: -1 + product of first n consecutive primes.
%C a(n) is congruent to -1 modulo the first n primes. - _Michael Engling_, Mar 31 2011
%C Named after the German mathematician Ernst Eduard Kummer (1810-1893). - _Amiram Eldar_, Jun 19 2021
%H T. D. Noe, <a href="/A057588/b057588.txt">Table of n, a(n) for n = 1..100</a>
%H E. E. Kummer, <a href="https://archive.org/details/monatsberichted01unkngoog/page/n837/">Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist</a>, Monatsber. Preuss. Akad. Wiss., Berlin 1878/9, pp. 777-778. [Collected Papers, II, pp. 669-670, Springer, Berlin-Heidelberg, 1975.] Cited in Mestrovic.
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>.
%H Robert G. Wilson v, <a href="/A038507/a038507.txt">Explicit factorizations</a>.
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F a(n) = A002110(n) - 1. - _Altug Alkan_, Oct 02 2015
%p seq(mul(ithprime(k), k=1..n) - 1, n=1..100); # _Muniru A Asiru_, Jan 19 2018
%t Table[Product[Prime[k], {k, 1, n}] - 1, {n, 1, 18}] (* _Artur Jasinski_, Jan 01 2007 *)
%t FoldList[Times,1,Prime[Range[20]]]-1 (* _Harvey P. Dale_, Apr 17 2011 *)
%t Table[ChineseRemainder[PadRight[{},n,-1],Prime[Range[n]]],{n,20}] (* _Harvey P. Dale_, Jul 01 2017 *)
%o (Haskell)
%o a057588 = (subtract 1) . product . (flip take a000040_list)
%o -- _Reinhard Zumkeller_, Mar 27 2013
%o (PARI) a(n) = prod(k=1, n, prime(k)) - 1; \\ _Michel Marcus_, Oct 02 2015
%o (Python)
%o from sympy import primorial
%o def A057588(n): return primorial(n)-1 # _Chai Wah Wu_, Feb 25 2023
%Y Cf. A002110, A006862, A057705 (primes), A000040.
%K nonn,easy,nice
%O 1,2
%A Mario Velucchi (mathchess(AT)velucchi.it), Oct 05 2000
%E More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000
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