A256580 Number of quadruples (x, x+1, x+2, x+3) with 1 < x < p-3 of consecutive integers whose product is 1 mod p.

0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 4, 0, 4, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 4, 2, 0, 2, 0, 0, 2, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4, 2, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 4, 0, 2, 2, 0, 0, 4, 4, 0, 4, 2, 0, 0

Offset

1,7

Comments

If "quadruples" is changed to "pairs" we get A086937 (for the counts) and A038872 (for the primes for which the count is nonzero).

Links

Table of n, a(n) for n=1..168.

Formula

|T| where T = {x|x*(x+1)*(x+2)*(x+3) == 1 mod p, p is prime, 1 < x < p-3}.

Example

p=7, x_1=2, 2*3*4*5 == 1 (mod 7), T={2}, |T|=1;
p=17, x_1=2, 2*3*4*5 == 1 (mod 17), x_2=12, 12*13*14*15 == 1 (mod 17), T={2,12}, |T|=2;
p=23, x_1=5, 5*6*7*8 == 1 (mod 23), x_2=15, 15*16*17*18 == 1 (mod 23), x_3=19, 19*20*21*22 == 1 (mod 23), T={5,15,19}, |T|=3.

Prog

(R)
library(numbers); IP <- vector(); t <- vector(); S <- vector(); IP <- c(Primes(1000)); for (j in 1:(length(IP))){for (i in 2:(IP[j]-4)){t[i-1] <-as.vector(mod((i*(i+1)*(i+2)*(i+3)), IP[j])); Z[j] <- sum(which(t==1)); S[j] <- length(which(t==1))}}; S

Crossrefs

Cf. A256567, A256572, A038872, A086937.

Sequence in context: A375519 A092241 A336124 * A213266 A182038 A128144

Adjacent sequences: A256577 A256578 A256579 * A256581 A256582 A256583

Keyword

nonn

Author

Marian Kraus, Apr 02 2015

Status

approved