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a(n) = (A364054(n) - A366470(n))/prime(n-1).
4

%I #28 Oct 28 2023 00:25:07

%S 1,2,2,0,1,0,1,2,2,1,2,0,1,0,1,0,1,0,1,0,1,2,0,1,0,1,2,3,0,2,0,1,0,1,

%T 0,2,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,

%U 0,1,0,1,0,1,0,1,0,1,2,0,1,2,2,1,2,1,2,0,1,2,0,2,0

%N a(n) = (A364054(n) - A366470(n))/prime(n-1).

%C a(29) = 3. When, if ever, does 4 appear?

%C Answer: a(28025) = 4. - _Michael De Vlieger_, Oct 26 2023

%H Michael De Vlieger, <a href="/A366475/b366475.txt">Table of n, a(n) for n = 2..65536</a>

%H Michael De Vlieger, <a href="/A366475/a366475.png">2048 X 2048 raster showing a(n)</a>, n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = 0, blue = 1, green = 2, gold = 3, red = 4.

%e n p(n-1) x y a(n) [x = A364054(n), y = A366470(n)]

%e 1 (1) 1 - - [a(n) = (x-y)/p(n-1)]

%e 2 2 3 1 1

%e 3 3 6 0 2

%e 4 5 11 1 2

%e 5 7 4 4 0

%e 6 11 15 4 1

%e 7 13 2 2 0

%e ...

%t nn = 2^20;

%t c[_] := False; m[_] := 0; a[1] = j = 1; c[0] = c[1] = True;

%t Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];

%t While[Set[k, p m[p] + r ]; c[k], m[p]++];

%t Set[{a[n], b[n], c[k], j}, {k, m[p], True, k}], {n, 2, nn}], n];

%t Array[b, nn-1, 2] (* _Michael De Vlieger_, Oct 26 2023 *)

%o (Python)

%o from itertools import count, islice

%o from sympy import nextprime

%o def A366475_gen(): # generator of terms

%o a, aset, p = 1, {0,1}, 1

%o while True:

%o p = nextprime(p)

%o b = a%p

%o for i in count(0):

%o if b not in aset:

%o aset.add(b)

%o a = b

%o break

%o b += p

%o yield i

%o A366475_list = list(islice(A366475_gen(),30)) # _Chai Wah Wu_, Oct 27 2023

%Y Cf. A364054, A366470, A366477 (records).

%K nonn

%O 2,2

%A _N. J. A. Sloane_, Oct 26 2023