A271564 Number of 6's found in the first differences of a reduced residue system modulo a primorial p#.

0, 0, 2, 14, 142, 1690, 26630, 470630, 10169950, 280323050, 8278462850, 293920842950, 11604850743850, 481192519512250, 21869408938627250, 1124832660535333750, 64590101883781223750, 3837395864206055401250, 250972362651045466681250, 17415757437491856599406250, 1243227958252662737649043750

Offset: 1

Author

Logan W. Wilbur, Apr 09 2016

Keywords

• nonn
• easy

Comments

Technically, the formula is undefined modulo 2# or 3#, but their values are listed as "0", since there are no 6's in the first differences of their reduced residue systems. For our purposes, by "6's", we mean n such that n,n+6 are relatively prime to the primorial modulus, while n+1,n+2,n+3,n+4,n+5 all share a factor (or factors) with p#. The values of this sequence are tied to actual distribution of sexy primes over N (conjecture).

Examples

			Modulo 5# (=30), there are (2*(5-2)-2*(5-3))=2 occurrences where n,n+6 are relatively prime, but n+1,n+2,n+3,n+4,n+5 share a factor with 30; they are n=1,n=23(mod30). Modulo 7# (=210), there are (2*(7-2)*(5-2)-2*(7-3)*(5-3))=30-16=14 such occurrences.
		

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 (d=2,4 values), A049296, A271565.

Programs

Mathematica

Table[2 Product[Prime@ k - 2, {k, 3, n}] - 2 Product[Prime@ k - 3, {k, 3, n}], {n, 21}] (* Michael De Vlieger, Apr 11 2016 *)

PARI

a(n) = 2*prod(k=3, n, prime(k)-2) - 2*prod(k=3, n, prime(k)-3); \\ Michel Marcus, Apr 10 2016

Formula

a(n) = 2*product(p-2)-2*product(p-3), where p runs over the primes greater than 3.

Extensions

Corrected and extended by Michel Marcus, Apr 10 2016