A253279 -id:A253279 - cp's OEIS frontend

A064413 EKG sequence (or ECG sequence): a(1) = 1; a(2) = 2; for n > 2, a(n) = smallest number not already used which shares a factor with a(n-1).

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, 27, 30, 25, 35, 28, 26, 13, 39, 36, 32, 34, 17, 51, 42, 38, 19, 57, 45, 40, 44, 46, 23, 69, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82

Offset: 1

Jonathan Ayres (Jonathan.ayres(AT)btinternet.com), Sep 30 2001

Locally, the graph looks like an EKG (American English) or ECG (British English).
Calculating the square of A064413 and plotting the results shows the EKG behavior even more dramatically - see A104125. - Parthasarathy Nambi, Jan 27 2005
Theorem: (1) Every number appears exactly once: this is a permutation of the positive numbers. - J. C. Lagarias, E. M. Rains, N. J. A. Sloane, Oct 03 2001
The permutation has cycles (1) (2) (3, 4, 6, 9, 10, 5) (..., 20, 18, 12, 7, 14, 13, 28, 26, ...) (8) ...
Theorem: (2) The primes appear in increasing order. - J. C. Lagarias, E. M. Rains, N. J. A. Sloane, Oct 03 2001
Theorem: (3) When an odd prime p appears it is immediately preceded by 2p and followed by 3p. - Conjectured by Lagarias-Rains-Sloane, proved by Hofman-Pilipczuk.
Theorem: (4) Let a'(n) be the same sequence but with all terms p and 3p (p prime) changed to 2p (see A256417). Then lim a'(n)/n = 1, i.e., a(n) ~ n except for the values p and 3p for p prime. - Conjectured by Lagarias-Rains-Sloane, proved by Hofman-Pilipczuk.
Conjecture: If a(n) != p, then almost everywhere a(n) > n. - Thomas Ordowski, Jan 23 2009
Conjecture: lim #(a_n > n) / n = 1, i.e., #(a_n > n) ~ n. - Thomas Ordowski, Jan 23 2009
Conjecture: A term p^2, p a prime, is immediately preceded by p*(p+1) and followed by p*(p+2). - Vladimir Baltic, Oct 03 2001. This is false, for example the sequence contains the 3 terms p*(p+2), p^2, p*(p+3) for p = 157. - Eric Rains
Theorem: If a(k) = 3p, then |{a(m) : a(m>k) < 3p}| = 3p - k. Proof: If a(k) = 3p, then all a(mk) > p and |{a(m) : a(m>k) < 3p}| = 3p - k. - Thomas Ordowski, Jan 22 2009
Let ...,a_i,...,2p,p,3p,...,a_j,... There does not exist a_i > 3p. There does not exist a_j < p. - Thomas Ordowski, Jan 20 2009
Let...,a,...,2p,p,3p,...,b,... All a<3p and b>p. #(a>2p) <= #(b<2p). - Thomas Ordowski, Jan 21 2009
If a(k)=3p then |{a(m):a(m>k)<3p}|=3p-k. - Thomas Ordowski, Jan 22 2009
GCD(a(n),n) = A247379(n). - Reinhard Zumkeller, Sep 16 2014
If the definition is changed to require that the GCD of successive terms be a prime power > 1, the sequence stays the same until a(578)=620, at which point a(579)=610 has GCD = 10 with the previous term. - N. J. A. Sloane, Mar 30 2015
From Michael De Vlieger, Dec 06 2021: (Start)
For prime p > 2, we have the chain {j : 2|j} -> 2p -> p -> 3p -> {k : 3|k}. The term j introducing 2p must be even, since 2p is an even squarefree semiprime proved by Hofman-Pilipczuk to introduce p itself. Hence no term a(i) such that p | a(i) exists in the sequence for i < n-1, where a(n) = p, leaving 2|j. Similarly, the term k following 3p must be divisible by 3 since the terms mp that are not coprime to p (thus implying p | mp) have m >= 4, thereby large compared to numbers k such that 3|k that belong to the cototient of 3p. For the chain {4, 6, 3, 9, 12}, the term 12 following 3p indeed is 4p, but p = 3; this is the only case of 4p following 3p in the sequence. As a consequence, for i > 1, A073734(A064955(i)-1) = 2 and A073734(A064955(i)+2) = 3.
For Fermat primes p, we have the chain {j : 2|j} -> 2^e-> {2p = 2^e + 2} -> {p = 2^(e-1) + 1} -> 3p -> {k : 3|k}.
     a(3) =      4 = 2^2,       a(5) =     3 = 2^1 + 1;
     a(8) =      8 = 2^3,      a(10) =     5 = 2^2 + 1;
    a(31) =     32 = 2^5,      a(33) =    17 = 2^4 + 1;
   a(485) =    512 = 2^9,     a(487) =   257 = 2^8 + 1;
a(127354) = 131072 = 2^17, a(127356) = 65537 = 2^16 + 1.
(End)

			a(2) = 2, a(3) = 4 (gcd(2,4) = 2), a(4) = 6 (gcd(4,6) = 2), a(5) = 3 (gcd(6,3) = 3), a(6) = 9 (6 already used so next number which shares a factor is 9 since gcd(3,9) = 3).

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Zak Seidov, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7.
Michael De Vlieger, Annotated plot of a(n) for n=1..120, showing prime p in red, 2p in blue, 3p in green, and other terms in gray.
Michael De Vlieger, Partially annotated log-log scatterplot of a(n) for n=1..1024, showing prime p in red, 2p in blue, 3p in green, and other terms in gray. This plot exhibits three quasi-linear striations, the densest contains both 2p and all "gray" terms outside of the first dozen or so terms in the sequence.
Michael De Vlieger, Table of n, a(n) for n = 1..262144.
Michael De Vlieger, Mathematica version of Eric Rains' C Code, 2021.
Diophante.fr, Les Récreations Mathématiques: E121. Une séquence cordiale.
Gordon Hamilton, The EKG Sequence and the Tree of Numbers
Gordon Hamilton, Untitled video related to previous video
Piotr Hofman and Marcin Pilipczuk, A few new facts about the EKG sequence, J. Integer Seqs., 11 (2008), Article 08.4.2.
James Keener, Mathematics of EKG [Refers to EKGs found in hospitals, included for comparison.]
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence, Exper. Math. 11 (2002), 437-446.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, Plot of a(1) to a(100), with successive points joined by lines.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, Terms 800 to 1000, with successive points joined by lines.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The first 1000 terms (represented by dots), successive points not joined.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The first 10000 terms (represented by dots), successive points not joined.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The sequence smoothed by replacing a(n)=p or 3p, p prime > 2, by a(n) = 2p.
Ivars Peterson, The EKG Sequence
E. M. Rains, C program
N. J. A. Sloane, Seven Staggering Sequences.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk).
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: Slides; Slides (an alternative source).
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 16.
Eric Weisstein's World of Mathematics, EKG Sequence
Index entries for sequences related to EKG sequence
Index entries for sequences that are permutations of the natural numbers

A073734 gives GCD's of successive terms.
See A064664 for the inverse permutation. See A064665-A064668 for the first two infinite cycles of this permutation. A064669 gives cycle representatives.
See A064421 for sequence giving term at which n appears.
See A064424, A074177 for records.
Cf. A064955 & A352194 (prime positions), A195376 (parity), A064957 (positions of odd terms), A064953 (positions of even terms), A064426 (first differences).
See A169857 and A119415 for the effect of changing the start.
Cf. A240024 (nonprime version).
Cf. A152458 (fixed points), A247379, A247383.
For other initial terms, see A169841, A169837, A169843, A169855, A169849.
A256417 is a smoothed version.
See also A255582, A256466, A257218, A257311-A257315, A257405, A253279 (two-dimensional analog).
See also A276127.

import Data.List (delete, genericIndex)
a064413 n = genericIndex a064413_list (n - 1)
a064413_list = 1 : f 2 [2..] where
   ekg x zs = f zs where
       f (y:ys) = if gcd x y > 1 then y : ekg y (delete y zs) else f ys
-- Reinhard Zumkeller, May 01 2014, Sep 17 2011

h := array(1..20000); a := array(1..10000); maxa := 300; maxn := 2*maxa; for n from 1 to maxn do h[n] := -1; od: a[1] := 2; h[2] := 1; c := 2; for n from 2 to maxa do for m from 2 to maxn do t1 := gcd(m,c); if t1 > 1 and h[m] = -1 then c := m; a[n] := c; h[c] := n; break; fi; od: od: ap := []: for n from 1 to maxa do ap := [op(ap),a[n]]; od: hp := []: for n from 2 to maxa do hp := [op(hp),h[n]]; od: convert(ap,list); convert(hp,list); # this is very crude!
N:= 1000: # to get terms before the first term > N
V:= Vector(N):
A[1]:= 1:
A[2]:= 2: V[2]:= 1:
for n from 3 do
  S:= {seq(seq(k*p,k=1..N/p),p=numtheory:-factorset(A[n-1]))};
  for s in sort(convert(S,list)) do
    if V[s] = 0 then
      A[n]:= s;
      break
    fi
  od;
  if V[s] = 1 then break fi;
  V[s]:= 1;
od:
seq(A[i],i=1..n-1); # Robert Israel, Jan 18 2016

maxN = 100; ekg = {1, 2}; unused = Range[3, maxN]; found = True; While[found, found = False; i = 0; While[ !found && i < Length[unused], i++; If[GCD[ekg[[-1]], unused[[i]]] > 1, found = True; AppendTo[ekg, unused[[i]]]; unused = Delete[unused, i]]]]; ekg (* Ayres *)
ekGrapher[s_List] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++ ]; Append[s, k]]; Nest[ekGrapher, {1, 2}, 71] (* Robert G. Wilson v, May 20 2009 *)

a1=1; a2=2; v=[1,2];
for(n=3,100,a3=if(n<0,0,t=1;while(vecmin(vector(length(v),i,abs(v[i]-t)))*(gcd(a2,t)-1)==0,t++);t);a2=a3;v=concat(v,a3););
a(n)=v[n];
/* Benoit Cloitre, Sep 23 2012 */

from math import gcd
A064413_list, l, s, b = [1,2], 2, 3, {}
for _ in range(10**5):
    i = s
    while True:
        if not i in b and gcd(i, l) > 1:
            A064413_list.append(i)
            l, b[i] = i, True
            while s in b:
                b.pop(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 08 2014

a(n) = smallest number not already used such that gcd(a(n), a(n-1)) > 1.

In Lagarias-Rains-Sloane (2002), it is conjectured that almost all a(n) satisfy the asymptotic formula a(n) = n (1+ 1/(3 log n)) + o(n/log n) as n -> oo and that the exceptional terms when the sequence is a prime or 3 times a prime p produce the spikes in the sequence. See the paper for a more precise statement of the conjecture. - N. J. A. Sloane, Mar 07 2015

More terms from Naohiro Nomoto, Sep 30 2001

Entry extensively revised by N. J. A. Sloane, Oct 10 2001

A354434 a(1) = 1; for n > 1, a(n) is the smallest unused square spiral number such that a(n) shares a factor with all existing numbers in its Moore neighborhood.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 18, 8, 10, 14, 16, 20, 22, 24, 15, 21, 27, 30, 33, 36, 39, 26, 28, 32, 34, 38, 40, 42, 44, 46, 48, 50, 54, 45, 35, 7, 63, 51, 57, 60, 66, 52, 72, 78, 84, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 75, 96, 100, 105, 49, 77, 91, 119, 102, 69, 81, 108, 92, 94, 98, 104

Offset: 1

Scott R. Shannon, May 28 2022

The sequence is conjectured to be a permutation of the positive integers, although it takes many terms for most primes to appear, e.g. a(1807) = 13, a(35156) = 179. The primes do not occur in their natural order. In the first 200000 terms the smallest unused prime is 181, while the smallest unused composite number is 11881, which is itself a prime power.

			The spiral begins
                                .
                                .
    7--35--45--54--50--48--46  82
    |                       |   |
   63  21--15--24--22--20  44  80
    |   |               |   |   |
   51  27   3---6---4  16  42  76
    |   |   |       |   |   |   |
   57  30   9   1---2  14  40  74
    |   |   |           |   |   |
   60  33  12--18---8--10  38  70
    |   |                   |   |
   66  36--39--26--28--32--34  68
    |                           |
   52--72--78--84--56--58--62--64
.
.
a(11) = 14 as the existing numbers in the Moore neighborhood when a(11) is being placed are 4,2,8,10, and 14 is the smallest unused number that shares a factor with all these numbers.

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Cf. A336946, A064413, A253279, A257112, A257339.

A336946 a(1) = 1; for n > 1, a(n) is the next square spiral number not already used such that a(n) shares a factor with a(n-1) and also with the adjacent number on the inner spiral arm if such a number exists.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 20, 14, 7, 28, 16, 18, 15, 21, 24, 22, 11, 33, 30, 25, 35, 40, 45, 36, 26, 42, 27, 63, 48, 32, 34, 50, 38, 54, 39, 51, 60, 44, 46, 66, 55, 65, 70, 49, 56, 52, 58, 72, 57, 76, 62, 78, 13, 117, 69, 75, 80, 64, 68, 74, 37, 148

Offset: 1

Scott R. Shannon, Aug 08 2020

nonn

This is a variation of the EKG sequence A064413 where the numbers are written on the square spiral such that each new number must share a common factor with not only the previous number but also with the adjacent inner spiral number, in one of the four axial directions, if such a number exists. This additional restriction causes the numbers to violate some of the patterns the numbers form in the standard EKG sequence, e.g., an odd prime p does not need to be preceded by 2p or followed by 3p, and the primes do not appear in increasing order.

For the first 100000 terms the smallest unseen number is 433.

			The spiral begins
                                .
                                .
   38--50--34--32--48--63--27  78
    |                       |   |
   54  15--18--16--28---7  42  62
    |   |               |   |   |
   39  21   3---6---4  14  26  76
    |   |   |       |   |   |   |
   51  24   9   1---2  20  36  57
    |   |   |           |   |   |
   60  22  12---8--10---5  45  72
    |   |                   |   |
   44  11--33--30--25--35--40  58
    |                           |
   46--66--55--65--70--49--56--52
.
a(1)-a(8) = 1,2,4,6,3,9,12,8. The adjacent inner spiral number is 1 which all numbers share a factor with so the numbers are the same as A064413(n).
a(9) = 10. This is the first number that must have a common factor with two numbers, the previous number a(8) = 8 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 10.
a(10) = 5. As this number is on the corner of a square spiral arm it only needs to share a divisor with a(9) = 10. The lowest unseen number satisfying this is 5.
a(11) = 20.  This number must have a common factor with the previous number a(10) = 5 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 20. This is also the first number to differ from  A064413 which only needs to find the lowest unused number sharing a factor with 5, which is 15.

Scott R. Shannon, Line graph of the first 1000 terms.

Cf. A064413, A073734, A253279, A257112.

A358153 Lexicographically earliest infinite sequence of distinct positive integers on a square spiral such that each number shares a factor with its four orthogonally nearest neighbors but shares no factor with its four diagonal next-nearest neighbors.

Original entry on oeis.org

6, 10, 35, 21, 77, 22, 143, 39, 65, 117, 12, 63, 18, 20, 24, 44, 26, 273, 30, 195, 36, 88, 42, 40, 48, 14, 455, 50, 175, 80, 55, 33, 385, 147, 539, 91, 105, 110, 847, 176, 1001, 28, 119, 51, 187, 153, 85, 459, 595, 15, 54, 351, 66, 189, 72, 99, 57, 95, 114, 100, 78, 220, 52, 60, 34, 833

Offset: 1

Scott R. Shannon, Nov 01 2022

nonn

This sequence is a 2D square lattice version of the Enots Wolley sequence A336957. Like that sequence, for this sequence to be infinite no number can be a prime or prime power - the first term must therefore be 6. Likewise, for each of its orthogonal nearest neighbors, a new number must have at least one prime factor that is not a factor of that neighbor. When choosing each new number a further test must also be performed against the previous number that is two squares ahead of the current number and on the edge of the spiral - this number is outside the eight nearest neighbors of the number being determined. See the examples below.

Like A336957 is it conjectured that all nonprime powers eventually appear. In the first 10000 terms the largest value is a(6975) = 3005315.

			The spiral begins:
.                                             .
                                              .
   105----91---539---147---385----33----55    99
    |                                   |     |
   110    26----44----24----20----18    80    72
    |     |                       |     |     |
   847   273    77----21----35    63   175   189
    |     |     |           |     |     |     |
   176    30    22     6----10    12    50    66
    |     |     |                 |     |     |
  1001   195   143----39----65---117   455   351
    |     |                             |     |
    28    36----88----42----40----48----14    54
    |                                         |
   119----51---187---153----85---459---595----15
.
.
a(5) = 22. The two orthogonal nearest neighbors to the square at (-1,0), relative to the starting central square 6, are 6 and 77. Therefore the new number at that position must share a factor with both of these numbers while containing a factor they do not have. Also the new number must have no factor in common with any diagonal next-nearest neighbors. Only one exists in this case, namely 21. The smallest unused number satisfying all of these conditions is 22.
a(6) = 143. For the square at (-1,-1) the orthogonal nearest neighbor is 22, while the diagonal next-nearest neighbor is 6. As 22 has 2 and 11 as factors, while 6 has 2 and 3, the new number can only be a multiple of 11, must have a factor other than 2 and 11 while not being a multiple of 2 or 3. The smallest unused number satisfying these conditions is 55. However this is where another check must be performed, namely against the number two squares ahead on the spiral from the current number - the number at (1,0) which in this case is 10. The reason this must be checked is that if 55 is chosen as the new number at (-1,-1) then that would force the next number at (0,-1) to be a multiple of 5 - only 5 and 11 are factors of 55 but 11 cannot be a factor of the new number at (0,-1) since it is an orthogonal neighbor to 22 with which it cannot share a factor. But forcing the number at (0,-1) to have 5 as a factor is not permitted since it would then share a factor with its diagonal neighbor 10 at (1,0). This is why the prime factors of the number two squares ahead on the spiral need to be checked when choosing the current number. Once a candidate number for the current square is chosen a list of the prime factors it has that are not factors of the previous number, 22 in this case, is created. This list is then compared against the factors of the number two squares ahead on the spiral, if such a number exists. If that number has as factors all the numbers in the list, then a new candidate number must be chosen else it would lead to the issue described above when the next new number is created. In this case choosing 55 leads to a list containing only 5, but that is a factor of 10, so 55 cannot be chosen. The next valid number that satisfies all the factor requirements and also has a factor not in either 22 or 10 is 11*13 = 143.

Scott R. Shannon, Image of the spiral up to n=10201 with even numbers shown in yellow. Zoom in to see the numbers and spiral line.
Scott R. Shannon, Image of the spiral up to n=10201 color coded with the minimum prime factor. The numbers with a minimum prime factor of 2, 3, 5, 7, 11, 13, 17, >=19 are shown as red, orange, yellow, green, blue, indigo, violet, dark grey respectively.

Cf. A336957, A336946, A253279, A257112, A335710, A336799, A346294.

cp's OEIS Frontend

A255999 Position of the n-th prime in A253279.

Views

Author

Keywords

Comments

Crossrefs

Formula

A064413 EKG sequence (or ECG sequence): a(1) = 1; a(2) = 2; for n > 2, a(n) = smallest number not already used which shares a factor with a(n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A354434 a(1) = 1; for n > 1, a(n) is the smallest unused square spiral number such that a(n) shares a factor with all existing numbers in its Moore neighborhood.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A336946 a(1) = 1; for n > 1, a(n) is the next square spiral number not already used such that a(n) shares a factor with a(n-1) and also with the adjacent number on the inner spiral arm if such a number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358153 Lexicographically earliest infinite sequence of distinct positive integers on a square spiral such that each number shares a factor with its four orthogonally nearest neighbors but shares no factor with its four diagonal next-nearest neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs