A256580 - OEIS

%I #29 Sep 12 2018 01:32:06

%S 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,

%T 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,

%U 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

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

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

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

%e p=7, x_1=2, 2*3*4*5 == 1 (mod 7), T={2}, |T|=1;

%e 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;

%e 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.

%o (R)

%o 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

%Y Cf. A256567, A256572, A038872, A086937.

%K nonn

%O 1,7

%A _Marian Kraus_, Apr 02 2015