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a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).
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%I #67 Mar 06 2024 11:58:44

%S 1,3,6,11,4,15,2,19,38,61,32,63,26,67,24,71,18,77,16,83,12,85,164,81,

%T 170,73,174,277,384,57,283,29,160,23,162,13,315,158,321,154,327,148,

%U 329,138,331,134,333,122,345,118,347,114,353,112,363,106,369,100,371,94,375,92,385

%N a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

%C 5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From _W. Edwin Clark_ via SeqFan.)

%C Comments from _N. J. A. Sloane_, Oct 22 2023 (Start)

%C It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.

%C It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.

%C The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious. It would be nice to know more.

%C Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.

%C (End)

%C a(28149) = 7. - _Chai Wah Wu_, Oct 22 2023

%C Comment from _N. J. A. Sloane_, Mar 05 2024 (Start):

%C At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.

%C The initial values of the inverse sequence are

%C n.....1..2..3..4..5..6....7.....8..9..10..11... . . .

%C index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

%H Chai Wah Wu, <a href="/A364054/b364054.txt">Table of n, a(n) for n = 1..10000</a>

%H N. J. A. Sloane, <a href="/A364054/a364054.pdf">Sketch showing the main features of the graph</a>

%H Michael De Vlieger, <a href="/A364054/a364054_2.png">Log log scatterplot of a(n)</a>, n = 1..2^20.

%H Michael De Vlieger, <a href="/A364054/a364054_3.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 = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).

%H Chai Wah Wu, <a href="/A364054/a364054_1.png">Graph of first 10^8 terms</a>

%e For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.

%e For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) = 6.

%e For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

%t a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* _Amiram Eldar_, Oct 20 2023 *)

%t nn = 2^20; 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], c[k], j}, {k, True, k}], {n, 2, nn}], n];

%t Array[a, nn] (* _Michael De Vlieger_, Oct 26 2023, fast, based on congruence, avoids search *)

%o (Python)

%o from itertools import count, islice

%o from sympy import nextprime

%o def A364054_gen(): # generator of terms

%o a, aset, p = 1, {0,1}, 2

%o while True:

%o yield a

%o for b in count(a%p,p):

%o if b not in aset:

%o aset.add(b)

%o a, p = b, nextprime(p)

%o break

%o A364054_list = list(islice(A364054_gen(),30)) # _Chai Wah Wu_, Oct 22 2023

%Y Cf. A006509, A125717.

%Y For a(n-1) (mod prime(n-1)) see A366470.

%Y Records: A368384, A368385.

%Y See also A366475, A366477.

%K nonn,look

%O 1,2

%A _Ali Sada_, Oct 19 2023.