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).

2, 3, 25, 301, 25201, 83161, 7207201, 49008961, 698377681, 2248776129601, 39594522567601, 2599263952084801, 160287943711896001, 4381203794791824001, 386203114510899285601, 130159869178331861668801

Offset: 1

Author

Mark Troll (mtroll(AT)u.washington.edu), Oct 22 2004

Keywords

• nonn

Examples

			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
		

Links

• Dimitri Papadopoulos, Table of n, a(n) for n = 1..167

Crossrefs

Cf. A099794, A099795, A099796.

Programs

Magma

/* By definition (slow): */
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 */
[p eq 2 select p else Modinv(p, Lcm([1..p-1]))*p: p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015

Mathematica

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 *)

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

Formula

a(n) = A099795(n) * A099796(n) + 1 = A099794(n) * prime(n).

log(a(n)) = prime(n) (approximately, empirical observation). - Dimitri Papadopoulos, Dec 27 2018

Extensions

Edited and extended by Ray Chandler, Oct 29 2004

Added "positive" to definition. - N. J. A. Sloane, Feb 10 2015