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A254939 a(n) = (A099795(n)^-1 mod p)*A099795(n), where p = prime(n).

%I A254939 #27 Feb 16 2025 08:33:24
%S A254939 1,4,36,120,2520,277200,5045040,183783600,4655851200,80313433200,
%T A254939 32607253879200,2743667504978400,58772246027695200,
%U A254939 5038384364010597600,56517528952814529600,34089489546705963770400,7391221142626702144764000
%N A254939 a(n) = (A099795(n)^-1 mod p)*A099795(n), where p = prime(n).
%C A254939 The sequence lists the smallest nonnegative solutions z to the system of congruences z == 1 (mod p), z == 0 (mod v(p-1)), where p is a prime and v(p-1) = lcm(1,...,p-1).
%H A254939 Umberto Cerruti, <a href="/A254939/a254939.pdf">Il Teorema Cinese dei Resti</a> (in Italian), 2015. The sequence is on page 21.
%H A254939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModularInverse.html">Modular Inverse</a>.
%F A254939 a(n) = A255010(n)*A099795(n).
%e A254939 5045040 is the seventh term of the sequence because the modular inverse of A099795(7) mod A000040(7) is 7 and 7*A099795(7) = 7*720720 = 5045040.
%p A254939 with(numtheory): P:=proc(q)  local a, n;  a:=[];
%p A254939 for n from 1 to q do a:=[op(a),n]; if isprime(n+1) then
%p A254939 print(lcm(op(a))*(lcm(op(a))^(-1) mod (n+1))); fi;
%p A254939 od; end: P(10^3); # _Paolo P. Lava_, Feb 16 2015
%t A254939 r[k_] := LCM @@ Range[k]; u[k_] := PowerMod[r[k - 1], -1, k] r[k - 1]; Table[u[Prime[n]], {n, 1, 20}]
%o A254939 (Magma) [Modinv(Lcm([1..p-1]),p)*Lcm([1..p-1]): p in PrimesUpTo(60)];
%o A254939 (PARI) a099795(n) = lcm(vector(prime(n)-1, k, k));
%o A254939 a(n) = {my(m = a099795(n)); m*lift(1/Mod(m, prime(n)));} \\ _Michel Marcus_, Feb 13 2015
%Y A254939 Cf. A000040, A056604, A099795, A254924, A255010.
%K A254939 nonn
%O A254939 1,2
%A A254939 _Bruno Berselli_, Feb 12 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)