A133243 -id:A133243 - cp's OEIS frontend

A134204 a(0)=2; for n > 0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1) + a(n) is a multiple of n. If no such prime exists, the sequence terminates.

2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37, 11, 67, 59, 61, 83, 53, 73, 79, 101, 109, 89, 233, 103, 47, 239, 139, 113, 293, 97, 151, 137, 127, 43, 167, 157, 509, 251, 373, 107, 467, 163, 181, 347, 193, 313, 439, 281, 307, 443, 271, 197, 227, 367, 733, 331, 353, 401, 71, 229

Offset: 0

Leroy Quet, Oct 14 2007

Is this sequence infinite and, if so, is it a permutation of the primes?
This sequence is infinite if and only if a(n-1) never divides n for any n.
This sequence exists for at least 800*10^6 terms (see A133242, A133243, A232992). - David Applegate, Nov 01 2007, Nov 15 2007
The plot of primes less than 10^6 shows an interesting crosshatch pattern. Why? [T. D. Noe, Jul 12 2009] See also the graph of A133244. - N. J. A. Sloane, Apr 06 2013
Entries A224221, A224222 are similar sequences which terminate after 20 or so steps, while A224223 and A224229 are similar sequences whose status is also unknown. - N. J. A. Sloane, Apr 05 2013
Empirically, the direction of hatchings is related to the parity of n, and each hatch corresponds to terms with the same value of Sum_{k=1..n} ((-1)^k * (a(k-1)+a(k))/k) (see colorized scatterplots in Links section). - Rémy Sigrist, Nov 07 2017

			The primes that don't occur among terms a(0) through a(6) form the sequence 11,23,29,31,... Of these, 23 is the smallest that when added to a(6)=19 gets a multiple of 7 -- 19+23 = 42 = 6*7. (19+11 = 30, which is not a multiple of 7.) So a(7) = 23.

Robert Israel and Reinhard Zumkeller, Table of n, a(n) for n = 0..100000 initial 1000 terms from Robert Israel
David Applegate, C++ Program [For output see A133242, A133243, A232992]
T. D. Noe, Graph of initial terms (out to 10^6)
N. J. A. Sloane, Eight Hateful Sequences, a short paper for the 8th Gathering for Gardner, May 2008.
Rémy Sigrist, Colored scatterplot of the sequence (where the color is a function of the parity of n)
Rémy Sigrist, Colored scatterplot of the sequence (where the color is a function of floor(a(n)/n))
Rémy Sigrist, Colored scatterplot of the sequence (where the color is a function of floor(2*a(n)/n))
Rémy Sigrist, Colored scatterplot of the sequence (where the color is a function of Sum_{k=1..n} ((-1)^k * (a(k-1)+a(k))/k))

Cf. A134205, A134206, A134207, A133242, A133243, A131261, A224221, A224222, A224223, A224229, A232992.
For records see A133244, A133245.
Cf. A162846 (where prime(n) occurs).

import Data.List (delete)
a134204 n = a134204_list !! n
a134204_list = 2 : f 1 2 (tail a000040_list) where
   f x q ps = p' : f (x + 1) p' (delete p' ps) where
     p' = head [p | p <- ps, mod (p + q) x == 0]
-- Reinhard Zumkeller, Jun 04 2012

aa = {a[0]=2, a[1]=3}; a[n_] := a[n] = (an = First[ Complement[ Prime[ Range[1 + PrimePi[ Max[aa]]]], aa]]; While[ Not[ FreeQ[aa, an] && Divisible[ a[n-1] + an, n]], an = NextPrime[an]]; AppendTo[aa, an]; an); Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 17 2012 *)
T. D. Noe, Apr 05 2013, provided the following information about how his plot (see link) was obtained: I computed 500000 points and then plotted up to y = 10^6. Here's the Mma code (which takes a while to run):
t = {2}; Do[k = Ceiling[t[[-1]]/n];
  While[p = k*n - t[[-1]]; ! PrimeQ[p] || MemberQ[t, p], k++];
  If[2 p < n, Print[{n, p, N[n/p]}]];
  AppendTo[t, p], {n, 500000}]
ListPlot[t, PlotRange -> {1, 1000000}, Frame -> True,
PlotStyle -> {PointSize[0.005]}, ImageSize -> 500,
PlotLabel -> "\nA134204(n)\n", GridLines -> Automatic]
With[{nn = 10^3}, Fold[Append[#1, SelectFirst[Prime@ Range[2, Ceiling@ Log2[nn] nn], Function[p, And[FreeQ[#1, p], Divisible[Last@ #1 + p, #2]]]]] &, {2}, Range@ nn]] (* Michael De Vlieger, Oct 16 2017 *)

A134204(n,show_all=1,a=2,used=[])={for(n=1,n, show_all & print1(a","); used=setunion(used,Set(a)); forstep(p=(-a)%n,9e19,n,isprime(p)||next; setsearch(used,p)&next; a=p;break));a} \\ M. F. Hasler, Mar 01 2013

More terms from Robert Israel, Oct 14 2007

A133242 Indices n such that A134204(n) < n.

Original entry on oeis.org

12, 201, 379, 474, 588, 868, 932, 1604, 1942, 2006, 3084, 4800, 7800, 9666, 9700, 10794, 10956, 11074, 11140, 11176, 14112, 16420, 16436, 16499, 17330, 17478, 18475, 20784, 21118, 21410, 22004, 22078, 22510

Offset: 1

Robert Israel, Oct 15 2007

nonn

			The first few exceptionally small terms in A134204 that give rise to this sequence and A133243 are b(12) = 11, b(201) = 173, b(379) = 257, b(474) = 263, b(588) = 571, b(868) = 631, b(932) = 887, ..., where b(i) = A134204(i).

David Applegate, Table of n, a(n) for n = 1..6148
David Applegate, C++ Program
David Applegate, Larger file giving first 49344 terms

Corrected and extended by David Applegate, Oct 15 2007

A232992 Let b(i) = A134204(i) and c(n) = A133242(n); a(n) is the number of primes p <= c(n) such that p is not in {b(0), b(1), ..., b(c(n)-1)}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 3, 6, 7, 6, 7, 7, 7, 6, 5, 7, 12, 11, 10, 10, 9, 10, 12, 11, 12, 11, 10, 9, 9, 8, 8, 8, 9, 8, 8, 8, 7, 10, 16, 16, 16, 19, 18, 17, 16, 15, 15, 16, 16, 17, 16, 15, 16, 16, 19, 19, 20, 20, 19, 18, 17, 16, 17, 20, 19, 20, 19, 18, 18, 19, 23, 24, 23, 25, 24, 25, 27, 26, 27, 27, 26, 25, 25

Offset: 1

N. J. A. Sloane, Dec 13 2013

nonn

Computed by David Applegate, Oct 2007.

Arises from studying the question of whether A134204 is an infinite sequence.

			Terms b(0) through b(12) of A134202 are (ignore the periods, which are just for alignment):
i:... 0, 1, 2, 3,. 4,. 5,. 6,. 7,. 8,. 9, 10, 11, 12
b(i): 2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37, 11
c(1) = 12 is the first i for which b(i)<i.
Then a(1) is the number of primes p <= 12 that are not in the set {b(0), ..., b(11)} = {2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37}.
Only p = 11 is missing, so a(1)=1.

David Applegate, Table of n, a(n) for n = 1..10000
David Applegate, C++ Program
David Applegate, Notes on programs and output
David Applegate, The first 106394 lines of output. The first 3 columns give the first 106394 terms of A133242, A133243 and A232992 (the present sequence), and establish that at least 800 million terms of A134204 exist.

Cf. A134204, A133242.

cp's OEIS Frontend

A134204 a(0)=2; for n > 0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1) + a(n) is a multiple of n. If no such prime exists, the sequence terminates.

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A133242 Indices n such that A134204(n) < n.

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A232992 Let b(i) = A134204(i) and c(n) = A133242(n); a(n) is the number of primes p <= c(n) such that p is not in {b(0), b(1), ..., b(c(n)-1)}.

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