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

1,4

Links

Table of n, a(n) for n=1..62.

Example

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.

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 *)

Prog

(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<j, prod(i:j), 0)
#loop: checks, if entry is congruent 1
#counts the "1"s in the matrix for each prime
for (i in 1:n)
NumTup[i] <- sum(mod(Piprod[, 1:(p[i]-1)], p[i])==1)
NumTup #vector of the counts of each prime

Crossrefs

Cf. A256567, A256580.

Sequence in context: A156816 A021383 A296456 * A256965 A191708 A368038

Adjacent sequences: A256589 A256590 A256591 * A256593 A256594 A256595

Keyword

nonn

Author

Marian Kraus, Apr 03 2015

Status

approved