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%I #18 Nov 14 2023 04:35:44
%S 0,0,0,2,28,394,6812,128810,2918020,83120450,2524575200,91589444450,
%T 3682730287600,155231331960250,7156139793803000,372520258834974250,
%U 21613446896458917500,1296556574981939521250,85520460088068245240000,5980843188551617897761250,430093937447553491544932500
%N Number of 8's found in the first differences of a reduced residue system modulo a primorial p#.
%C Technically, the formula is undefined modulo 2# or 3#, but I have listed their values as "0", since there are no 8's in the first differences of their reduced residue systems. For our purposes, by "8's", we mean n such that n,n+8 are relatively prime to the primorial modulus, while n+1,n+2,n+3,n+4,n+5,n+6,n+7 all share a factor (or factors) with p#.
%H Steven Brown, <a href="https://arxiv.org/abs/2311.06873">Distance between consecutive elements of the multiplicative group of integers modulo n</a>, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.
%F a(n) = product(p-2) - 2*product(p-3) + product(p-4), where p runs through the primes > 3 and <= prime(n).
%e Modulo 5# (=30), there are (5-2)-2*(5-3)+(5-4)=0 occurrences where n, n+8 are relatively prime but n+1, n+2, n+3, n+4, n+5, n+6, n+7 share a factor with 30.
%e Modulo 7# (=210), there are (7-2)(5-2)-2*(7-3)(5-3)+(7-4)(5-4)=15-16+3=2 such occurrences; i.e when n=89,113 (mod210).
%t Table[Product[Prime@ k - 2, {k, 3, n}] - 2 Product[Prime@ k - 3, {k, 3, n}] + Product[Prime@ k - 4, {k, 3, n}], {n, 21}] (* _Michael De Vlieger_, Apr 11 2016 *)
%o (PARI) a(n) = prod(k=3, n, prime(k)-2) - 2*prod(k=3, n, prime(k)-3) + prod(k=3, n, prime(k)-4); \\ _Michel Marcus_, Apr 11 2016
%Y Cf. A049296, A059861, A271564.
%K nonn,easy
%O 1,4
%A _Logan W. Wilbur_, Apr 10 2016
%E More terms from _Michel Marcus_, Apr 11 2016
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