A341805 -id:A341805 - cp's OEIS frontend

A079276 Multiplicative inverse in the finite field F(prime(n)) of the product of the first n-1 primes modulo prime(n).

1, 2, 1, 4, 1, 3, 15, 18, 20, 12, 18, 27, 7, 5, 43, 2, 4, 10, 38, 3, 60, 20, 53, 62, 52, 83, 11, 30, 27, 49, 113, 63, 79, 25, 81, 143, 80, 121, 53, 142, 81, 52, 81, 150, 136, 40, 176, 114, 167, 138, 84, 46, 239, 213, 137, 4, 122, 136, 255, 141, 273, 30, 22, 25, 179, 9, 43, 12

Offset: 1

Valentin F. Schmid (v_schmid(AT)hotmail.com), Feb 07 2003

nonn

a(n)=1 if and only if n-1 is in A341805. - Jeppe Stig Nielsen, Feb 20 2021

			a(6)=3 because 2*3*5*7*11 = 2310, 2310 == 9 (mod 13) and 9*(9^(-1)) == 9*3 == 1 (mod 13).

Eric Weisstein's World of Mathematics, Primorial

Cf. A062347, A002110, A000040, A240673, A341805.

a := n -> (1/mul(ithprime(j), j=1..n-1)) mod ithprime(n);
seq(a(n), n=1..68); # Peter Luschny, Apr 13 2014

a[n_] := Module[{i}, Return[PowerMod[Product[Prime[i], {i, 1, n - 1}], -1, Prime[n]]]; ];

a(1) = 1; for n>1, a(n) = ( prime(n-1)# (mod prime(n)) )^(-1), where prime(i) is the i-th prime number, prime(i)# is the product of first i primes, x^(-1) is the multiplicative inverse in the finite field F(prime(n)).

A341804 Primes p dividing (the product of the primes less than p)-1.

Original entry on oeis.org

2, 5, 11, 176078293

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

The initial term 2 is included because the empty product minus 1 (which gives zero) is divisible by 2.

			The prime 11 is included because 2*3*5*7-1 is divisible by 11. Therefore, the last factor of the product, namely 7, is in A341812.

Cf. A058233, A066735, A341805, A341812.

t=1;forprime(p=2,,((t-1)%p==0)&&print1(p,", ");t*=p)

A338543 Numbers k such that 1 + Product_{i

Original entry on oeis.org

1, 2, 8, 233, 431

Offset: 1

Robert Israel, Nov 01 2020

Numbers k such that A062347(k-1) == -1 (mod prime(k)).

			a(3)=8 is a term because 1 + 2*3*5*7*11*13*17 = 510511 is divisible by prime(8)=19.

Mathematics StackExchange, Is 510511 the smallest euclid number e_n that divides by the next prime p.

Cf. A002110, A006862, A081618, A062347, A341805.

select(t -> 1+mul(ithprime(i),i=1..t-1) mod ithprime(t)=0, [$1..1000]);

isok(n) = ((1+vecprod(primes(n-1))) % prime(n)) == 0; \\ Michel Marcus, Nov 03 2020

a(n) = A081618(n)+1 for n >= 2.

A341812 Primes p such that p#-1 is divisible by the next prime after p.

Original entry on oeis.org

3, 7, 176078267

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

Here p# means A034386(p).

The next prime after p is then in A341804.

			The prime 7 is a member of this sequence because 2*3*5*7-1=209 is divisible by 11, the next prime immediately following 7.
11 is NOT a member of this sequence because 2*3*5*7*11-1=2309 is not divisible by 13 (leaves a remainder of 8), the next prime after 11.

Carlos Rivera, Puzzle 117. Certain p#+1 values, contains message from Wilfred Whiteside mentioning this sequence.

Cf. A057588, A058233, A341804, A341805.

isok(p) = if (isprime(p), my(a=Mod(1,nextprime(1+p))); forprime(q=2,p,a*=q); (a == 1)); \\ Michel Marcus, Mar 03 2021

cp's OEIS Frontend

A079276 Multiplicative inverse in the finite field F(prime(n)) of the product of the first n-1 primes modulo prime(n).

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A341804 Primes p dividing (the product of the primes less than p)-1.

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A338543 Numbers k such that 1 + Product_{i

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A341812 Primes p such that p#-1 is divisible by the next prime after p.

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PARI