A329152 a(n) = Sum_{i=1..n-1} Sum_{j=1..i-1} [1 == i*j (mod n)], where [] is the Iverson bracket.
0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 5, 2, 2, 2, 7, 2, 8, 2, 4, 4, 10, 0, 9, 5, 8, 4, 13, 2, 14, 6, 8, 7, 10, 4, 17, 8, 10, 4, 19, 4, 20, 8, 10, 10, 22, 4, 20, 9, 14, 10, 25, 8, 18, 8, 16, 13, 28, 4, 29, 14, 16, 14, 22, 8, 32, 14, 20, 10, 34, 8, 35, 17, 18, 16, 28, 10, 38, 12
Offset: 1
Examples
a(1)=0 because there is no solution to k*1 + 1 = a*b, 1 < a < b < n, 1 <= k < n. a(5)=1 because 1 == 3*2 (mod 5). a(7)=2 because 1 == 4*2 == 5*3 (mod 7). a(11)=4 because 1 == 4*3 == 6*2 == 8*7 == 9*5 (mod 11). a(13)=5 because 1 == 7*2 == 8*5 == 9*3 == 10*4 = 11*6 (mod 13). a(32)=6 because 1 == 11*3 == 13*5 == 23*7 == 25*9 == 27*19 == 29*21 (mod 32).
Links
- David Broadhurst, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
Array[Sum[Sum[Boole[Mod[i j, #] == 1], {j, i - 1}], {i, # - 1}] &, 80] (* Michael De Vlieger, Mar 15 2020 *) -
PARI
a(n) = {my(x=0); for (i = 1, n - 1, for (ii = 1, i - 1, if(1 == ((ii*i) % n), x++))); return(x)} -
PARI
a(n)=sum(i=2,n-2,gcd(i,n)==1&&(1/i)%n>i); \\ David Broadhurst, Feb 20 2025
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PARI
a(n)=eulerphi(n)/2-2^(#znstar(n)[3]-1); \\ David Broadhurst, Feb 21 2025
Comments