A256592 - OEIS

%I #8 Apr 04 2015 09:10:15

%S 0,0,1,2,6,3,8,7,13,15,13,11,13,22,18,25,36,31,34,53,42,38,38,40,55,

%T 47,41,37,77,59,62,67,66,63,55,84,74,78,90,74,90,92,85,108,100,117,98,

%U 104,136,114,118,118,141,112,118,115,122,138,132,129,115,152

%N Let p = prime(n); a(n) = number of pairs (x,i) with i >= 2 and 2 <= x <= p-i such that x*(x+1)*(x+2)*...*(x+i-1) == 1 mod p.

%e prime(1)=2: There is no such product

%e => a(1)=0;

%e prime(2)=3: There is no such product

%e => a(2)=0;

%e prime(3)=5: 2*3=6==1 mod 5

%e => i=1, x=2; a(3)=1;

%e prime(4)=7: 4*5*6==1 mod 7; 2*3*4*5==1 mod 7

%e => a(3)=2;

%e prime(5)=11: 3*4==1 mod 11; 7*8==1 mod 11; 5*6*7==1 mod 11; 3*4*5*6*7==1 mod 11; 6*7*8*9*10==1 mod 11; 2*3*4*5*6*7*8*9==1 mod 11

%e => x in {3,7,5,3,6,2}

%e => a(5)=6.

%t f[n_] := Block[{r = Range[2, Prime[n] - 1]}, Sum[Length@ Select[Times @@@ Partition[r, k, 1], Mod[#, Prime@ n] == 1 &], {k, 2, Prime@ n}]]; Array[f, 72] (* _Michael De Vlieger_, Apr 03 2015 *)

%o (R)

%o library(numbers)

%o p <- vector()

%o n <- vector()

%o NumTup <- vector()

%o p <- Primes(m)

%o n <- length(p)

%o m <- 17 #all primes will be checked up to this number

%o Piprod <- matrix(0,m,m) #Matrix with zeros

%o #loop: every ordered combination of products

%o for (i in 2:m)

%o for (j in 2:m)

%o   Piprod[j,i] <- ifelse(i<j,prod(i:j),0)

%o #loop: checks, if entry is congruent 1

%o #counts the "1"s in the matrix for each prime

%o for (i in 1:n)

%o NumTup[i] <- sum(mod(Piprod[,1:(p[i]-1)],p[i])==1)

%o NumTup #vector of the counts of each prime

%Y Cf. A256567, A256580.

%K nonn

%O 1,4

%A _Marian Kraus_, Apr 03 2015