cp's OEIS Frontend

A364054 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)).

%I A364054 #69 Jul 09 2025 05:02:38
%S A364054 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 A364054 170,73,174,277,384,57,283,29,160,23,162,13,315,158,321,154,327,148,
%U A364054 329,138,331,134,333,122,345,118,347,114,353,112,363,106,369,100,371,94,375,92,385
%N A364054 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 A364054 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 A364054 Comments from _N. J. A. Sloane_, Oct 22 2023 (Start)
%C A364054 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 A364054 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 A364054 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 A364054 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 A364054 (End)
%C A364054 a(28149) = 7. - _Chai Wah Wu_, Oct 22 2023
%C A364054 Comment from _N. J. A. Sloane_, Mar 05 2024 (Start):
%C A364054 At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
%C A364054 The initial values of the inverse sequence are
%C A364054 n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
%C A364054 index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)
%H A364054 Chai Wah Wu, <a href="/A364054/b364054.txt">Table of n, a(n) for n = 1..10000</a>
%H A364054 N. J. A. Sloane, <a href="/A364054/a364054.pdf">Sketch showing the main features of the graph</a>
%H A364054 Michael De Vlieger, <a href="/A364054/a364054_2.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A364054 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 A364054 Chai Wah Wu, <a href="/A364054/a364054_1.png">Graph of first 10^8 terms</a>
%e A364054 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 A364054 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 A364054 For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.
%t A364054 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 A364054 nn = 2^20; c[_] := False; m[_] := 0; a[1] = j = 1; c[0] = c[1] = True;
%t A364054   Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
%t A364054     While[Set[k, p m[p] + r ]; c[k], m[p]++];
%t A364054     Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}], n];
%t A364054 Array[a, nn] (* _Michael De Vlieger_, Oct 26 2023, fast, based on congruence, avoids search *)
%o A364054 (Python)
%o A364054 from itertools import count, islice
%o A364054 from sympy import nextprime
%o A364054 def A364054_gen(): # generator of terms
%o A364054     a, aset, p = 1, {0,1}, 2
%o A364054     while True:
%o A364054         yield a
%o A364054         for b in count(a%p,p):
%o A364054             if b not in aset:
%o A364054                 aset.add(b)
%o A364054                 a, p = b, nextprime(p)
%o A364054                 break
%o A364054 A364054_list = list(islice(A364054_gen(),30)) # _Chai Wah Wu_, Oct 22 2023
%Y A364054 Cf. A006509, A125717.
%Y A364054 For a(n-1) (mod prime(n-1)) see A366470.
%Y A364054 Records: A368384, A368385.
%Y A364054 See also A366475, A366477.
%K A364054 nonn,look
%O A364054 1,2
%A A364054 _Ali Sada_, Oct 19 2023