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%I #21 Nov 14 2023 04:34:38
%S 0,0,2,14,142,1690,26630,470630,10169950,280323050,8278462850,
%T 293920842950,11604850743850,481192519512250,21869408938627250,
%U 1124832660535333750,64590101883781223750,3837395864206055401250,250972362651045466681250,17415757437491856599406250,1243227958252662737649043750
%N Number of 6'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 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).
%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) = 2*product(p-2)-2*product(p-3), where p runs over the primes greater than 3.
%e 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.
%t 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 *)
%o (PARI) a(n) = 2*prod(k=3, n, prime(k)-2) - 2*prod(k=3, n, prime(k)-3); \\ _Michel Marcus_, Apr 10 2016
%Y Cf. A059861 (d=2,4 values), A049296, A271565.
%K nonn,easy
%O 1,3
%A _Logan W. Wilbur_, Apr 09 2016
%E Corrected and extended by _Michel Marcus_, Apr 10 2016
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