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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.

%I A256580 #29 Sep 12 2018 01:32:06
%S A256580 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 A256580 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 A256580 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 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.
%C A256580 If "quadruples" is changed to "pairs" we get A086937 (for the counts) and A038872 (for the primes for which the count is nonzero).
%F A256580 |T| where T = {x|x*(x+1)*(x+2)*(x+3) == 1 mod p, p is prime, 1 < x < p-3}.
%e A256580 p=7, x_1=2, 2*3*4*5 == 1 (mod 7), T={2}, |T|=1;
%e A256580 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 A256580 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 A256580 (R)
%o A256580 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 A256580 Cf. A256567, A256572, A038872, A086937.
%K A256580 nonn
%O A256580 1,7
%A A256580 _Marian Kraus_, Apr 02 2015