1,4

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#.

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.

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

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.

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).

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 *)

(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

Cf. A049296, A059861, A271564.

Sequence in context: A061629 A230130 A011944 * A241365 A240771 A012745

Adjacent sequences: A271562 A271563 A271564 * A271566 A271567 A271568

nonn,easy

Logan W. Wilbur, Apr 10 2016

More terms from Michel Marcus, Apr 11 2016

approved