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A094998 a(n) is smallest positive integer multiple of n-th prime, say k*prime(n), such that k*prime(n) == 1 (mod j) for each integer j with 1 < j < prime(n).

%I A094998 #43 Sep 08 2022 08:45:13
%S A094998 2,3,25,301,25201,83161,7207201,49008961,698377681,2248776129601,
%T A094998 39594522567601,2599263952084801,160287943711896001,
%U A094998 4381203794791824001,386203114510899285601,130159869178331861668801
%N A094998 a(n) is smallest positive integer multiple of n-th prime, say k*prime(n), such that k*prime(n) == 1 (mod j) for each integer j with 1 < j < prime(n).
%H A094998 Dimitri Papadopoulos, <a href="/A094998/b094998.txt">Table of n, a(n) for n = 1..167</a>
%F A094998 a(n) = A099795(n) * A099796(n) + 1 = A099794(n) * prime(n).
%F A094998 log(a(n)) = prime(n) (approximately, empirical observation). - _Dimitri Papadopoulos_, Dec 27 2018
%e A094998 For n = 1 there are no such j, so the condition is vacuously satisfied and we can take k=1, getting a(1)=2. - _N. J. A. Sloane_, Feb 10 2015
%t A094998 a[1] = 2; a[2] = 3; a[n_] := Module[{p, m, r, r0, r1}, p = Prime[n]; m = LCM @@ Range[2, p - 1]; r = Reduce[k > 0 && p*k + m*j == 1, {k, j}, Integers]; r0 = r /. C[_] -> 0; r1 = r /. C[_] -> 1; If[r0 === False, r1[[1, 2]], Min[r0[[1, 2]], r1[[1, 2]]]]*p]; Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Feb 09 2015 *)
%o A094998 (Magma) /* By definition (slow): */
%o A094998 S:=[]; for n in [1..9] do k:=1; while not forall{j: j in [2..NthPrime(n)-1] | IsOne(k*NthPrime(n) mod j)} do k:=k+1; end while; Append(~S, k*NthPrime(n)); end for; S; /* or */
%o A094998 [p eq 2 select p else Modinv(p, Lcm([1..p-1]))*p: p in PrimesUpTo(60)]; // _Bruno Berselli_, Feb 08 2015
%o A094998 (PARI) a(n) = {p=prime(n); k=1; for(n=2, p-1, k=lcm(k,n)); for(j=1, p, if((j*k+1)/p==ceil((j*k+1)/p), t=j*k+1; break())); return(t);} \\ _Dimitri Papadopoulos_, Dec 28 2018
%Y A094998 Cf. A099794, A099795, A099796.
%K A094998 nonn
%O A094998 1,1
%A A094998 Mark Troll (mtroll(AT)u.washington.edu), Oct 22 2004
%E A094998 Edited and extended by _Ray Chandler_, Oct 29 2004
%E A094998 Added "positive" to definition. - _N. J. A. Sloane_, Feb 10 2015