cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A098550 The Yellowstone permutation: a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17, 18, 85, 24, 55, 34, 65, 36, 91, 30, 49, 38, 63, 19, 42, 95, 44, 57, 40, 69, 50, 23, 48, 115, 52, 75, 46, 81, 56, 87, 62, 29, 31, 58, 93, 64, 99, 68, 77, 54, 119, 60
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 14 2004

Keywords

Comments

For n > 3, gcd(a(n), a(n-1)) = 1 and gcd(a(n), a(n-2)) > 1. (This is just a restatement of the definition.)
This is now known to be a permutation of the natural numbers: see the 2015 article by Applegate, Havermann, Selcoe, Shevelev, Sloane, and Zumkeller.
From N. J. A. Sloane, Nov 28 2014: (Start)
Some of the known properties (but see the above-mentioned article for a fuller treatment):
1. The sequence is infinite. Proof: We can always take a(n) = a(n-2)*p, where p is a prime that is larger than any prime dividing a(1), ..., a(n-1). QED
2. At least one-third of the terms are composite. Proof: The sequence cannot contain three consecutive primes. So at least one term in three is composite. QED
3. For any prime p, there is a term that is divisible by p. Proof: Suppose not. (i) No prime q > p can divide any term. For if a(n)=kq is the first multiple of q to appear, then we could have used kp < kq instead, a contradiction. So every term a(n) is a product of primes < p. (ii) Choose N such that a(n) > p^2 for all n > N. For n > N, let a(n)=bg, a(n+1)=c, a(n+2)=dg, where g=gcd(a(n),a(n+2)). Let q be the largest prime factor of g. We know q < p, so qp < p^2 < dg, so we could have used qp instead of dg, a contradiction. QED
3a. Let a(n_p) be the first term that is divisible by p (this is A251541). Then a(n_p) = q*p where q is a prime less than p. If p < r are primes then n_p < n_r. Proof: Immediate consequences of the definition.
4. (From David Applegate, Nov 27 2014) There are infinitely many even terms. Proof:
Suppose not. Then let 2x be the maximum even entry. Because the sequence is infinite, there exists an N such that for any n > N, a(n) is odd, and a(n) > x^2.
In addition, there must be some n > N such that a(n) < a(n+2). For that n, let g = gcd(a(n),a(n+2)), a(n) = bg, a(n+1)=c, a(n+2)=dg, with all of b,c,d,g relatively prime, and odd.
Since dg > bg, d > b >= 1, so d >= 3. Also, g >= 3.
Since a(n) = bg > x^2, one of b or g is > x.
Case 1: b > x. Then 2b > 2x, so 2b has not yet occurred in the sequence. And gcd(bg,2b)=b > x > 1, gcd(2b,c)=1, and since g >= 3, 2b < bg < dg. So a(n+2) should have been 2b instead of dg.
Case 2: g > x. Then 2g > 2x, so 2g has not yet occurred in the sequence. And gcd(bg,2g)=g > 1, gcd(2g,c)=1, and since d >= 3, 2g < dg. So a(n+2) should have been 2g instead of dg.
In either case, we derive a contradiction. QED
Conjectures:
5. For any prime p > 97, the first time we see p, it is in the subsequence a(n) = 2b, a(n+2) = 2p, a(n+4) = p for some n, b, where n is about 2.14*p and gcd(b,p)=1.
6. The value of |{k=1,..,n: a(k)<=k}|/n tends to 1/2. - Jon Perry, Nov 22 2014 [Comment edited by N. J. A. Sloane, Nov 23 2014 and Dec 26 2014]
7. Based on the first 250000 terms, I conjectured on Nov 30 2014 that a(n)/n <= (Pi/2)*log n.
8. The primes in the sequence appear in their natural order. This conjecture is very plausible but as yet there is no proof. - N. J. A. Sloane, Jan 29 2015
(End)
The only fixed points seem to be {1, 2, 3, 4, 12, 50, 86} - see A251411. Checked up to n=10^4. - L. Edson Jeffery, Nov 30 2014. No further terms up to 10^5 - M. F. Hasler, Dec 01 2014; up to 250000 - Reinhard Zumkeller; up to 300000 (see graph) - Hans Havermann, Dec 01 2014; up to 10^6 - Chai Wah Wu, Dec 06 2014; up to 10^8 - David Applegate, Dec 08 2014.
From N. J. A. Sloane, Dec 04 2014: (Start)
The first 250000 points lie on about 8 roughly straight lines, whose slopes are approximately 0.467, 0.957, 1.15, 1.43, 2.40, 3.38, 5.25 and 6.20.
The first six lines seem well-established, but the two lines with highest slope at present are rather sparse. Presumably as the number of points increases, there will be more and more lines of ever-increasing slopes.
These lines can be seen in the Havermann link. See the "slopes" link for a list of the first 250000 terms sorted according to slope (the four columns in the table give n, a(n), the slope a(n)/n, and the number of divisors of a(n), respectively).
The primes (with two divisors) all lie on the lowest line, and the lines of slopes 1.43 and higher essentially consist of the products of two primes (with four divisors).
(End)
The eight roughly straight lines mentioned above are actually curves. A good fit for the "line" with slope ~= 1.15 is a(n)~=n(1+1.0/log(n/24.2)), and a good fit for the other "lines" is a(n)~= (c/2)*n(1-0.5/log(n/3.67)), for c = 1,2,3,5,7,11,13. The first of these curves consists of most of the odd terms in the sequence. The second family consists of the primes (c=1), even terms (c=2), and c*prime (c=3,5,7,11,13,...). This functional form for the fit is motivated by the observed pattern (after the first 204 terms) of alternating even and odd terms, except for the sequence pattern 2*p, odd, p, even, q*p when reaching a prime (with q a prime < p). - Jon E. Schoenfield and David Applegate, Dec 15 2014
For a generalization, see the sequence of monomials of primes in the comment in A247225. - Vladimir Shevelev, Jan 19 2015
From Vladimir Shevelev, Feb 24 2015: (Start)
Let P be prime. Denote by S_P*P the first multiple of P appearing in the sequence. Then
1) For P >= 5, S_P is prime.
Indeed, let
a(n-2)=v, a(n-1)=w, a(n)=S_P*P. (*)
Note that gcd(v,P)=1. Therefore, by the definition of the sequence, S_P*P should be the smallest number such that gcd(v,S_P) > 1.
So S_P is the smallest prime factor of v.
2) The first multiples of all primes appear in the natural order.
Suppose not. Then there is a pair of primes P < Q such that S_Q*Q appears earlier than S_P*P. Let
a(m-2)=v_1, a(m-1)=w_1, a(m)=S_Q*Q. (**)
Then, as in (*), S_Q is the smallest prime factor of v_1. But this does not depend on Q. So S_Q*P is a smaller candidate in (**), a contradiction.
3) S_P < P.
Indeed, from (*) it follows that the first multiple of S_P appears earlier than the first multiple of P. So, by 2), S_P < P.
(End)
For any given set S of primes, the subsequence consisting of numbers whose prime factors are exactly the primes in S appears in increasing order. For example, if S = {2,3}, 6 appears first, in due course followed by 12, 18, 24, 36, 48, 54, 72, etc. The smallest numbers in each subsequence (i.e., those that appear first) are the squarefree numbers A005117(n), n > 1. - Bob Selcoe, Mar 06 2015

Links

Crossrefs

Cf. A098548, A098551, A249943 (first time all 1..n appear), A251553.
The inverse permutation is in A098551.
A098552(n) = a(a(n)).
A251102(n) = GCD(a(n+2),a(n)).
Cf. A251101 (smallest prime factor), A251103 (largest prime factor), A251138 (number of distinct prime factors), A251140 (total number of prime factors), A251045 (squarefree part), A251089 (squarefree kernel), A250127 and A251415 (records for a(n)/n), A251411 (fixed points), A248647 (records).
Cf. also A251412 (trajectory of 11), A251556 (finite cycles), A251413 and A251414 (variant involving odd numbers), A249357 ("Fibonacci" variant).
Smallest missing numbers: A251416, A251417, A251546-A251552, A247253. See also A251557, A241558, A251559.
Indices of some pertinent subsequences: A251237 (even numbers), A251238 (odd numbers), A251391 (squarefree), A251541 and A251239 (primes), A251240 (squares of primes), A251241 (prime powers), A251393 (powers of 2), A251392 (nonprimes), A253297 (primes p for which some multiple k*p > 2*p precedes p).
Three arrays concerning the occurrences of multiples of primes: A251637, A251715, A251716.
Two sequences related to the numbers which immediately follow a prime: A253048, A253049. Seven sequences related to the numbers that appear two steps after primes: A251542, A251543, A251544, A251545, A253052, A253053, A253054. See also A253055 and A253056.
Other starting values: A251554, A251555.
See also A251756, A253297, A251662, A253572, A253573, A253591, A253593, A253588, A253590, A253609, A252865, A252867, A252868, A247225, A247942, A254003, A254077, A254669, A254670, A255509 (version with a priority for appearance of the primes), A255615, A255617, A256189, A256224, A256368, A256461.
See also A064413 (EKG sequence), A255582, A121216 (similar sequences), A257112 (two-dimensional analog).
See also A253765 and A253766 (bisections), A250299 (parity), A253768 (partial sums).
See A336957 for a variation.

Programs

  • Haskell
    import Data.List (delete)
a098550 n = a098550_list !! (n-1)
a098550_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v ws = g ws where
     g (x:xs) = if gcd x u > 1 && gcd x v == 1
                   then x : f v x (delete x ws) else g xs
-- Reinhard Zumkeller, Nov 21 2014
  • Maple
    N:= 10^4: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N,datatype=integer[4]):
for n from 1 to 3 do A[n]:= n: od:
for n from 4 do
  for k from 4 to N do
    if B[k] = 0 and igcd(k,A[n-1]) = 1 and igcd(k,A[n-2]) > 1 then
       A[n]:= k;
       B[k]:= 1;
       break
    fi
  od:
  if k > N then break fi
od:
seq(A[i],i=1..n-1); # Robert Israel, Nov 21 2014
  • Mathematica
    f[lst_List] := Block[{k = 4}, While[ GCD[ lst[[-2]], k] == 1 || GCD[ lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]]; Nest[f, {1, 2, 3}, 68] (* Robert G. Wilson v, Nov 21 2014 *)
NN = Range[4, 1000]; a098550 = {1, 2, 3}; g = {-1}; While[g[[1]] != 0, g = Flatten[{FirstPosition[NN, v_ /; GCD[a098550[[-1]], v] == 1 && GCD[a098550[[-2]], v] > 1, 0]}]; If[g[[1]] != 0, d = NN[[g]]; a098550 = Flatten[Append[a098550, d[[1]]]]; NN = Delete[NN, g[[1]]]]]; Table[a098550[[n]], {n, 71}] (* L. Edson Jeffery, Jan 01 2015 *)
  • PARI
    a(n, show=1, a=3, o=2, u=[])={n<3&&return(n); show&&print1("1, 2"); for(i=4,n, show&&print1(","a); u=setunion(u, Set(a)); while(#u>1 && u[2]==u[1]+1, u=vecextract(u,"^1")); for(k=u[1]+1, 9e9, gcd(k,o)>1||next; setsearch(u,k)&&next; gcd(k,a)==1||next; o=a; a=k; break)); a} \\ Replace "show" by "a+1==i" in the main loop to print only fixed points. - M. F. Hasler, Dec 01 2014
  • Python
    from math import gcd
A098550_list, l1, l2, s, b = [1,2,3], 3, 2, 4, {}
for _ in range(1,10**6):
    i = s
    while True:
        if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
            A098550_list.append(i)
            l2, l1, b[i] = l1, i, 1
            while s in b:
                b.pop(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 04 2014

A251237 Indices of even numbers in A098550.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 63, 65, 67, 69, 71, 73, 75, 77, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 125, 128, 130, 133, 135
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 02 2014

Keywords

Comments

A098550(a(n)) mod 2 = 0;
a(n+1) > a(n) + 1;
there are infinitely many even terms in A098550, for proof: see comment #4 in A098550.

Links

Crossrefs

Cf. A098550, A005843, A251238, A251393.
First row of array A251716.

Programs

  • Haskell
    a251237 n = a251237_list !! (n-1)
a251237_list = filter (even . a098550) [1..]
  • Mathematica
    f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
Position[Nest[f, {1, 2, 3}, 140], ?EvenQ] // Flatten (* _Jean-François Alcover, Oct 01 2018, after Robert G. Wilson v in A098550 *)

A251541 Index of first term in A098550 that is divisible by the n-th prime.

Original entry on oeis.org

2, 3, 7, 8, 20, 21, 28, 41, 49, 59, 60, 77, 85, 86, 99, 112, 125, 130, 140, 151, 156, 165, 173, 192, 202, 213, 220, 231, 236, 245, 272, 281, 294, 299, 322, 327, 336, 353, 362, 373, 384, 391, 412, 421, 428, 433, 450, 477, 488, 495, 504, 521, 526, 539, 548, 569, 580, 587
Offset: 1

Views

Author

N. J. A. Sloane, Dec 15 2014

Keywords

Comments

It is known that every prime divides some term of A098550, so the sequence exists.
Conjectures. 1: Equals first column of A251716. 2: For n > 4, a(n) = A251239(n) - 2. - Reinhard Zumkeller, Dec 15 2014

Links

Crossrefs

Cf. A098550, A251239, A251544, A251716.
Cf. A253297.

A251637 Square array read by antidiagonals containing in row n the multiples of prime(n) in A098550 in order of appearance.

Original entry on oeis.org

2, 3, 4, 15, 9, 8, 14, 5, 15, 14, 22, 35, 25, 6, 6, 39, 11, 7, 35, 12, 12, 51, 13, 33, 21, 10, 21, 16, 38, 17, 26, 55, 28, 20, 27, 10, 69, 19, 85, 65, 44, 91, 45, 39, 20, 87, 23, 95, 34, 91, 99, 49, 85, 33, 22, 62, 29, 115, 57, 68, 52, 77, 63, 55, 45, 26, 74
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 07 2014

Keywords

Comments

T(n,k) = A251715(n,k)*A000040(n); A251715(n,k) = T(n,k)/A000040(n);
T(n,k) = A098550(A251716(n,k)); A251716(n,k) = A098551(T(n,k));
T(n,1) = A251618(n); for n > 4: T(n,2) = A000040(n);
conjecture: A098550 is a permutation of the positive integers iff A001221(n) = number of rows containing n.
A251541 = first column, and A251544 = third column for row numbers > 4. - Reinhard Zumkeller, Dec 16 2014

Examples

			.   n   p |  first 14 multiples of p = prime(n) in A098550, n = 1..25
.  -------+-------------------------------------------------------------
.   1   2 |   2  4   8  14   6  12  16  10   20  22   26   28   32   18
.   2   3 |   3  9  15   6  12  21  27  39   33  45   51   18   24   36
.   3   5 |  15  5  25  35  10  20  45  85   55  65   30   95   40   50
.   4   7 |  14 35   7  21  28  91  49  63   42  56   77  119  133  161
.   5  11 |  22 11  33  55  44  99  77  66   88 165  143  121  187  110
.   6  13 |  39 13  26  65  91  52 117  78  104 195  143  130  156  221
.   7  17 |  51 17  85  34  68 119 153 102  187 136  170  255  221  204
.   8  19 |  38 19  95  57 133  76 171 114  152 209  247  190  285  228
.   9  23 |  69 23 115  46 161  92 138 207  184 253  299  345  230  276
.  10  29 |  87 29  58 145 203 116 174 261  232 319  377  290  435  348
.  11  31 |  62 31  93 155 124 217 279 186  341 403  248  465  310  372
.  12  37 |  74 37 111 185 148 259 222 333  296 407  555  370  629  481
.  13  41 | 123 41  82 205 164 287 246 369  451 328  410  533  615  492
.  14  43 |  86 43 129 215 172 301 387 258  473 344  430  645  559  516
.  15  47 |  94 47 329 141 235 188 282 423  517 376  470  611  705  564
.  16  53 | 106 53 265 159 212 371 318 477  424 583  689  530  795  636
.  17  59 | 118 59 177 295 236 413 354 531  649 472  767  590  885 1003
.  18  61 | 122 61 427 183 305 244 366 549  671 488  793  610  915  732
.  19  67 | 201 67 335 134 268 469 603 402  536 737  871  670 1005  804
.  20  71 | 142 71 213 355 284 497 426 639  568 781  710 1065  923  852
.  21  73 | 146 73 365 219 292 511 438 657  584 803  730  949 1095  876
.  22  79 | 158 79 237 395 316 553 474 711  632 869 1027  790 1185  948
.  23  83 | 249 83 581 166 415 332 498 747  913 664 1079  830 1245  996
.  24  89 | 178 89 267 445 356 623 534 801  712 979 1157  890 1335 1068
.  25  97 | 291 97 679 194 485 388 582 873 1067 776  970 1261 1455 1164 .
.  ---------------------------------------------------------------------
See also A251715 for a table with T(n,k)/p and A251716 for a table of indices of T(n,k) within A098550.

Links

Crossrefs

Cf. A098550, A000040, A251618 (first column), A001221, A251715, A251716.
Cf. A251541, A251544.

Programs

  • Haskell
    when seen as table read by rows:
a251637 n k = a251637_tabl !! (n-1) !! (k-1)
a251637_row n= a251637_tabl !! (n-1)
a251637_tabl = adias $ map
   (\k -> filter
     ((== 0) . flip mod (fromInteger $ a000040 k)) a098550_list) [1..]
  • Mathematica
    rows = 25; (* f = A098550 *) Clear[f, row]; f[n_ /; n <= 3] := n; f[n_] := f[n] = Module[{k}, For[k=4, GCD[f[n-2], k] == 1 || GCD[f[n-1], k]>1 || MemberQ[Array[f, n-1], k], k++]; k]; row[n_] := row[n] = Module[{k, cnt}, Reap[For[k=1; cnt=0, cnt <= rows-n, k++, If[Divisible[f[k], Prime[n]], cnt++; Sow[f[k]]]]][[2, 1]]]; A251637 = Table[row[n-k+1][[k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 17 2014 *)

A251553 Numbers k such that A098550(k) is a multiple of 3.

Original entry on oeis.org

3, 5, 7, 10, 12, 17, 19, 21, 24, 26, 28, 31, 33, 37, 39, 42, 44, 47, 49, 52, 55, 57, 59, 64, 66, 69, 71, 73, 75, 78, 81, 83, 85, 90, 92, 95, 97, 100, 102, 104, 107, 109, 111, 113, 115, 117, 120, 123, 129, 131, 133, 136, 138, 140, 143, 145, 149, 152, 155, 157, 159, 162, 164, 169, 171, 173, 176, 178
Offset: 1

Views

Author

N. J. A. Sloane, Dec 18 2014

Keywords

Comments

This sequence is now known to be infinite.

Links

Crossrefs

Cf. A098550.
Second row of array A251716.

Programs

  • Haskell
    a251553 n = a251553_list !! (n-1)
a251553_list = filter ((== 0) . flip mod 3 . a098550) [1..]
-- Reinhard Zumkeller, Dec 19 2014
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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