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A069561 Start of a run of n consecutive positive numbers divisible respectively by first n primes.

%I A069561 #31 Dec 14 2021 20:12:50
%S A069561 2,2,8,158,788,788,210998,5316098,34415168,703693778,194794490678,
%T A069561 5208806743928,138782093170508,5006786309605868,253579251611336438,
%U A069561 12551374903381164638,142908008812141343558,77053322014980646906358
%N A069561 Start of a run of n consecutive positive numbers divisible respectively by first n primes.
%C A069561 It is evident that from a(3) onwards terms must be congruent to 8 mod p(3)#, where p(n)# is the n-th primorial (A002110). In fact the sequence for A069561(n) == k (mod p(n)#) for k: 2, 2, 8, 788, 788, 210988, etc. This follows from the Chinese Remainder Theorem.
%H A069561 Charles R Greathouse IV, <a href="/A069561/b069561.txt">Table of n, a(n) for n = 1..350</a>
%F A069561 log a(n) << n log n. - _Charles R Greathouse IV_, Jun 20 2015
%e A069561 a(5) = 788 as 788, 789, 790, 791 and 792 are divisible by 2, 3, 5, 7, and 11 respectively.
%t A069561 f[n_] := ChineseRemainder[-Range[0, n - 1], Prime[Range[n]]]; Array[f, 17, 2] (* _Robert G. Wilson v_, Jan 13 2012 *)
%t A069561 (* This code uses memoization in calculating the coeff for the primorial assoc'ed with a(n) value to generate a(n+1), producing 1000 terms in under one second (on a 2017 Costco Dell 64-bit Windows 10 machine)*)
%t A069561 q[1] =0; q[2] =0;
%t A069561 q[n_]:= (ModularInverse[Product[Prime[i], {i,1,n-1}], Prime[n]] * Mod[Prime[n]-n+1-g[n-1], Prime[n]])  // Mod[#, Prime[n]]&;
%t A069561 g[1] =2; g[2] =2;
%t A069561 g[r_] :=g[r]= g[r-1] + q[r] * Product[Prime[i], {i,1,r-1}];
%t A069561 Array[g, 1000]
%t A069561 (* _Christopher Lamb_, Oct 19 2021 *)
%o A069561 (PARI) a(n)=lift(chinese(vector(max(n,2),k,Mod(1-k,prime(k))))) \\ _Charles R Greathouse IV_, Jun 20 2015
%Y A069561 Cf. A072562.
%K A069561 nonn
%O A069561 1,1
%A A069561 _Amarnath Murthy_, Mar 22 2002
%E A069561 More terms to a(15) from _Sascha Kurz_, Mar 23 2002
%E A069561 Edited and extended by _Robert G. Wilson v_, Aug 09 2002