A062347 -id:A062347 - cp's OEIS frontend

A128856 Index of first appearance of n in A062347.

Original entry on oeis.org

0, 1, 71, 21, 45, 12, 64, 6, 5, 280, 11, 121, 638185, 62, 8, 76, 9, 7, 10

Offset: 1

Zak Seidov, Apr 16 2007, revised Apr 17 2007

nonn

a(20) > pi(10^7)=664579

a(n) = smallest m such that A062347(m)=n.

More terms from Max Alekseyev, Apr 20 2007

A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st prime.

Original entry on oeis.org

1, 1, 2, 2, 8, 36, 288, 1920, 2880, 120960, 362880, 6386688, 34836480, 217728000, 3881779200, 275904921600, 1785411403776, 28217548800000, 608662978560000, 3492203839488000, 964122158039040000, 2224367550332928000, 1314079960596480000000, 3758268687305932800000

Offset: 1

Hans Boelens (h.p.m.boelens(AT)pl.hanze.nl), Feb 02 2005

nonn

			Prime(6) = 13, 13 mod 2 = 1, 13 mod 3 = 1, 13 mod 5 = 3, 13 mod 7 = 6, 13 mod 11 = 2 so a(6) = 1*1*3*6*2 = 36.

Robert Israel, Table of n, a(n) for n = 1..413

Cf. A033955, A062347.

f:= proc(n) local p,i;
  p:= ithprime(n);
  mul(p mod ithprime(i),i=1..n-1)
end proc:
map(f, [$1..25]); # Robert Israel, Jan 12 2021

f[n_] := Times @@ Mod[ Prime[n], Table[ Prime[i], {i, n - 1}]]; Table[ f[n], {n, 22}] (* Robert G. Wilson v, Feb 04 2005 *)
Join[{0},Table[Times@@Mod[Prime[n],Prime[Range[n-1]]],{n,2,30}]] (* Harvey P. Dale, May 16 2019 *)

a(n) = my(pr = 1, pn = prime(n)); forprime (q=1, precprime(pn-1), pr *= (pn % q)); pr; \\ Michel Marcus, Jan 12 2021

More terms from Robert G. Wilson v, Feb 04 2005

a(1) (an empty product, therefore 1 by standard convention) corrected by N. J. A. Sloane, Jan 11 2021

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

Original entry on oeis.org

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)).

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.

A066909 (product of primes < n that do not divide n) (mod n).

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 2, 1, 7, 1, 1, 1, 9, 3, 7, 7, 8, 17, 18, 9, 16, 15, 15, 1, 24, 1, 26, 13, 17, 1, 19, 9, 14, 23, 3, 23, 11, 31, 19, 1, 6, 37, 26, 21, 7, 45, 35, 35, 25, 41, 2, 17, 27, 1, 31, 43, 50, 41, 15, 49, 55, 9, 10, 51, 28, 37, 30, 13, 26, 47, 24, 13, 28, 27, 73, 73, 76, 43, 4

Offset: 1

Leroy Quet, Jan 22 2002

			a(8) = (3 * 5 * 7) (mod 8) = 1 because 3, 5 and 7 are the primes < 8 that do not divide 8.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A062347, A066838.

a(n) = prod(i=1, n-1, if (isprime(i) && (n%i) , i, 1)) % n; \\ Michel Marcus, May 20 2014

a(n) = A066838(n) modulo n. - Michel Marcus, May 20 2014

a(prime(n)) = A062347(n-1). - Michel Marcus, May 20 2014

A126147 a(n) = floor((Product_{k=1..n-1} prime(k))/prime(n)).

Original entry on oeis.org

0, 0, 1, 4, 19, 177, 1766, 26868, 421725, 7692857, 208699781, 5420553787, 180993613044, 7075587523888, 278356624078085, 11601694011103611, 552358618257458385, 31520661477937912115, 1750572856110551805720

Offset: 1

Leroy Quet, Mar 07 2007

nonn

Every distinct prime dividing ((Product_{k=1..n-1} prime(k)) (mod prime(n))) also divides a(n).

Let Pn(n) = A002110(n) denote the primorial function. The number of natural numbers < Pn(n) that have prime(n+1) as a prime factor is equal to a(n). For example 19 numbers < Pn(4) = 210 have 11 as a prime factor. - Jamie Morken, Sep 18 2018

Muniru A Asiru, Table of n, a(n) for n = 1..185

Cf. A062347, A002110.

seq(floor(mul(ithprime(k),k=1..n-1)/ithprime(n)),n=1..20); # Muniru A Asiru, Sep 21 2018

f[n_] := Floor[ Product[ Prime@k, {k, n - 1}] / Prime@n]; Array[f, 19] (* Robert G. Wilson v, Mar 07 2007 *)

More terms from Robert G. Wilson v, Mar 07 2007

A340712 Primes p such that p divides (2 + product of primes < p).

Original entry on oeis.org

557, 248137, 4085791, 519807973

Offset: 1

Martin Ehrenstein, Jan 16 2021

			557 is in the sequence because 2 + A034386(557 - 1) = 557 * 4627335992...5904782776 (220 digits).

Norman Luhn, Prime Curios! 557

Cf. A338543, A062347, A034386.

from sympy import nextprime
def aupto(limit):
  p, psharp = 3, 2
  while p <= limit:
    if (psharp+2)%p == 0: print(p, end=", ")
    psharp, p = psharp*p, nextprime(p)
aupto(500000) # Michael S. Branicky, Mar 24 2021

A371598 a(n) = (Product_{i=1..n} Fibonacci(i)) mod Fibonacci(n + 1).

Original entry on oeis.org

0, 1, 2, 1, 6, 6, 12, 2, 15, 16, 0, 49, 299, 220, 882, 252, 2176, 166, 495, 5720, 5251, 6065, 28224, 41650, 106947, 113288, 256737, 173841, 26840, 25379, 444150, 347278, 1834953, 8709610, 4046544, 2653673, 31127545, 47532000, 50717205, 147239197, 97769672, 37543458

Offset: 1

Adnan Baysal, Mar 29 2024

nonn

			a(1) = 0 since A000045(1) = A000045(2) = 1 and 1 mod 1 = 0.
a(2) = (1 * 1) mod 2 = 1.
a(3) = (1 * 1 * 2) mod 3 = 2.
a(4) = (1 * 1 * 2 * 3) mod 5 = 1.

Cf. A000045, A003266, A062347, A333599.

a[n_] := Mod[Fibonorial[n], Fibonacci[n + 1]]; Array[a, 50] (* Amiram Eldar, Mar 29 2024 *)

a(n) = my(f=fibonacci(n+1)); lift(prod(k=1, n, Mod(fibonacci(k), f))); \\ Michel Marcus, Apr 03 2024

from sympy import fibonacci
def a(n):
    a_n = 1
    mod = fibonacci(n + 1)
    for i in range(1, n + 1):
        a_n = (a_n * fibonacci(i)) % mod
    return a_n

a(n) = A003266(n) mod A000045(n+1).

cp's OEIS Frontend

A128856 Index of first appearance of n in A062347.

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A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st prime.

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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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A338543 Numbers k such that 1 + Product_{i

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A066909 (product of primes < n that do not divide n) (mod n).

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A126147 a(n) = floor((Product_{k=1..n-1} prime(k))/prime(n)).

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Maple

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A340712 Primes p such that p divides (2 + product of primes < p).

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Python

A371598 a(n) = (Product_{i=1..n} Fibonacci(i)) mod Fibonacci(n + 1).

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