A285298 Number of 10's found in the first differences of a reduced residue system modulo a primorial p#.

0, 0, 0, 2, 30, 438, 7734, 148530, 3401790, 97648950, 2985436650, 108861586050, 4396116829650, 186022750845750, 8604610718954250, 449203003036037250, 26126835342151293750, 1570919774837171508750, 103827535054074567986250, 7274630596396103444253750

Offset: 1

Author

Andrew Fuchs, Apr 16 2017

Keywords

• nonn

Comments

Technically, the formula is undefined modulo 2# or 3#, but I have listed their values as "0", since there are no 10's in the first differences of their reduced residue systems. For our purposes, by "10's", we mean n such that n,n+10 are relatively prime to the primorial modulus, while n+1,n+2,n+3,n+4,n+5,n+6,n+7,n+8,n+9 all share a factor (or factors) with p#.

Links

• Steven Brown, Distance between consecutive elements of the multiplicative group of integers modulo n, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.

Crossrefs

Cf. A059861, A271564, A271565.

Programs

Mathematica

Table[4*Product[-2 + Prime[z], {z, 4, i}] -
   6*Product[-3 + Prime[z], {z, 4, i}] +
   2*Product[-4 + Prime[z], {z, 4, i}], {i, 20}]

Formula

a(n) = 4*product(p-2) - 6*product(p-3) + 2*product(p-4), where p runs through the primes > 5 and <= prime(n).