A256592 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.
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, 47, 41, 37, 77, 59, 62, 67, 66, 63, 55, 84, 74, 78, 90, 74, 90, 92, 85, 108, 100, 117, 98, 104, 136, 114, 118, 118, 141, 112, 118, 115, 122, 138, 132, 129, 115, 152
Offset: 1
Keywords
Examples
prime(1)=2: There is no such product
=> a(1)=0;
prime(2)=3: There is no such product
=> a(2)=0;
prime(3)=5: 2*3=6==1 mod 5
=> i=1, x=2; a(3)=1;
prime(4)=7: 4*5*6==1 mod 7; 2*3*4*5==1 mod 7
=> a(3)=2;
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
=> x in {3,7,5,3,6,2}
=> a(5)=6.
Programs
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Mathematica
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 *) -
R
library(numbers) p <- vector() n <- vector() NumTup <- vector() p <- Primes(m) n <- length(p) m <- 17 #all primes will be checked up to this number Piprod <- matrix(0,m,m) #Matrix with zeros #loop: every ordered combination of products for (i in 2:m) for (j in 2:m) Piprod[j,i] <- ifelse(i