A254939 -id:A254939 - cp's OEIS frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

1, 0, 0, 1, 130, 1329, 1707670, 27502484, 209927657739, 130904517147542068, 3673771932850374193, 69623451054783204822486486, 3724616892817543661693877073170, 149157913707716515940392007441860, 12429106799179771738076359013310638297

Offset: 1

Bruno Berselli, Feb 12 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

Let theta(p) be the smallest nonnegative solution z to the system of congruences z == 0 (mod p), z == 1 (mod v(p-1)), where p is a prime and v(p-1) = lcm(1,...,p-1). Theta(p) is unique mod lcm(p, v(p-1)), therefore it is unique mod v(p). Since both (p-1)!+1 and theta(p) are solutions to these congruences, ((p-1)!+1 - theta(p))/v(p) is always an integer. The sequence lists the values of this ratio (assuming theta(2)=0 and p=prime(n)).

			For n=5, a(5) = (A060371(5) - A094998(5))/A056604(5) = (3628801 - 25201)/27720 = 130.

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Il Teorema Cinese dei Resti (in Italian), 2015. The sequence is on page 21.

Cf. A000040, A056604, A060371, A094998, A254939, A255010.

[(Factorial(p-1)+1-Modinv(p,Lcm([1..p-1]))*p)/Lcm([1..p]): p in PrimesUpTo(50)];

with(numtheory): P:=proc(q)  local a,j,k,ok,n;  print(1); a:=[1];
for n from 3 to q do k:=0; a:=[op(a),n]; if isprime(n) then ok:=0;  while ok=0 do ok:=1;
k:=k+1; for j from 2 to n-1 do if not (k*n mod j)=1 then ok:=0; break; fi; od; od;
print((((n-1)!+1)-k*n)/lcm(op(a))); fi; od; end: P(100); # Paolo P. Lava, Feb 16 2015

r[k_] := LCM @@ Range[k]; s[k_] := PowerMod[k, -1, r[k - 1]] k; w[k_] := ((k - 1)! + 1 - s[k])/r[k]; Table[w[Prime[n]], {n, 1, 20}]

A255010 a(n) = A099795(n)^-1 mod prime(n).

Original entry on oeis.org

1, 2, 3, 2, 1, 10, 7, 15, 20, 1, 14, 19, 11, 23, 6, 11, 45, 42, 37, 34, 10, 29, 76, 77, 14, 71, 12, 88, 40, 22, 30, 75, 115, 59, 110, 14, 113, 154, 13, 154, 142, 40, 50, 25, 71, 16, 11, 18, 91, 174, 138, 35, 115, 38, 27, 195, 206, 113, 75, 119, 181, 111, 203

Offset: 1

Bruno Berselli, Feb 13 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

By the definition, a(n)*A099795(n) == 1 (mod prime(n)).

a(n) is 1 with the primes 2, 11, 29, 787, 15773 (see A178629).

Robert Israel, Table of n, a(n) for n = 1..10000
Umberto Cerruti, Il Teorema Cinese dei Resti (in Italian), 2015. The sequence is on page 21.
Eric Weisstein's World of Mathematics, Modular Inverse

Cf. A000040, A099795, A178629, A254924, A254939.

[Modinv(Lcm([1..p-1]),p): p in PrimesUpTo(400)];

with(numtheory): P:=proc(q)  local a, n;  a:=[];
for n from 1 to q do a:=[op(a),n]; if isprime(n+1) then print(lcm(op(a))^(-1) mod (n+1)); fi;
od; end: P(10^3); # Paolo P. Lava, Feb 16 2015

r[k_] := LCM @@ Range[k]; t[k_] := PowerMod[r[k - 1], -1, k]; Table[t[Prime[n]], {n, 1, 70}]

a(n) = lift(1/Mod(lcm(vector(prime(n)-1, k, k)), prime(n))); \\ Michel Marcus, Feb 13 2015

[inverse_mod(lcm([1..p-1]),p) for p in primes(400)]

a(n) = A254939(n)/A099795(n).

cp's OEIS Frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

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