A257218 -id:A257218 - 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

A257120 Position of first occurrence of n in A256918.

Original entry on oeis.org

1, 4, 3, 5, 7, 11, 29, 13, 10, 15, 122, 19, 200, 31, 24, 18, 299, 22, 824, 16, 33, 124, 945, 41, 26, 202, 21, 37, 3093, 45, 7059, 40, 126, 301, 53, 66, 13527, 826, 204, 42, 28658, 36, 40810, 130, 57, 947, 46657, 64, 80, 44, 303, 208, 57449, 49, 133, 38, 828

Offset: 1

Reinhard Zumkeller, Apr 25 2015

nonn

A256918(a(n)) = n and A256918(m) != n for m < a(n);
GCD(A257218(a(n)),A257218(a(n)+1)) = n;
if A257218 is a permutation, then also this sequence is a permutation;
a(n) < A257475(n); a(n) = A257475(n) - A257478(n).
a(n) >= A257122(n)-1. - Ivan Neretin, May 02 2015

			See A257478.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..72

Cf. A256918, A257475, A257478, A257218.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a257120 = (+ 1) . fromJust . (`elemIndex` a256918_list)

a(41)-a(57) from Hiroaki Yamanouchi, May 03 2015

A257475 Position of second and last occurrence of n in A256918.

Original entry on oeis.org

2, 6, 9, 12, 8, 14, 30, 17, 20, 28, 123, 23, 201, 32, 25, 39, 300, 46, 825, 43, 34, 125, 946, 51, 27, 203, 47, 52, 3094, 56, 7060, 62, 127, 302, 54, 71, 13528, 827, 205, 61, 28659, 79, 40811, 132, 58, 948, 46658, 65, 81, 99, 304, 210, 57450, 69, 134, 77, 829

Offset: 1

Reinhard Zumkeller, Apr 25 2015

nonn

GCD(A257218(a(n)),A257218(a(n)-1)) = n;

a(n) > A257120(n); a(n) = A257120 + A257478(n).

			Cf. A257478.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..72

Cf. A256918, A257120, A257478, A257218.

a257475 n = f 1 a256918_list where
   f i (u:us) = (if u == n then g else f) (i + 1) us
   g j (v:vs) =  if v == n then j else g  (j + 1) vs

a(38)-a(57) from Hiroaki Yamanouchi, May 03 2015

A257478 Distance of positions of first and last occurrence of n in A256918.

Original entry on oeis.org

1, 2, 6, 7, 1, 3, 1, 4, 10, 13, 1, 4, 1, 1, 1, 21, 1, 24, 1, 27, 1, 1, 1, 10, 1, 1, 26, 15, 1, 11, 1, 22, 1, 1, 1, 5, 1, 1, 1, 19, 1, 43, 1, 2, 1, 1, 1, 1, 1, 55, 1, 2, 1, 20, 1, 39, 1, 1, 1, 13, 1, 1, 32, 42, 1, 11, 1, 2, 1, 67, 1, 26

Offset: 1

Reinhard Zumkeller, Apr 25 2015

nonn

a(n) = A257475(n) - A257120(n).

			Let w(n) = A257218(n),
u(n) = A257120(n), xx'(n) = (w(u(n)),w(u(n)+1)),
v(n) = A257475(n), yy'(n) = (w(v(n)),w(v(n)+1)):
. ----+------+------+------++--------------+------------+---------+
.   n | a(n) | u(n) | v(n) ||       xx'(n) |     yy'(n) | ... gcd |
. ----+------+------+------++--------------+------------+---------+
.   1 |    1 |    1 |    2 ||     (1,   2) |   (2,   3) |       1 |
.   2 |    2 |    4 |    6 ||     (6,   4) |   (8,  10) |       2 |
.   3 |    6 |    3 |    9 ||     (3,   6) |  (15,   9) |       3 |
.   4 |    7 |    5 |   12 ||     (4,   8) |  (12,  16) |       4 |
.   5 |    1 |    7 |    8 ||    (10,   5) |   (5,  15) |       5 |
.   6 |    3 |   11 |   14 ||    (18,  12) |  (24,  30) |       6 |
.   7 |    1 |   29 |   30 ||    (70,   7) |   (7,  14) |       7 |
.   8 |    4 |   13 |   17 ||    (16,  24) |  (40,  32) |       8 |
.   9 |   10 |   10 |   20 ||     (9,  18) |  (36,  27) |       9 |
.  10 |   13 |   15 |   28 ||    (30,  20) |  (50,  70) |      10 |
.  11 |    1 |  122 |  123 ||   (660,  11) |  (11,  22) |      11 |
.  12 |    4 |   19 |   23 ||    (48,  36) |  (72,  60) |      12 |
.  13 |    1 |  200 |  201 ||  (1092,  13) |  (13,  26) |      13 |
.  14 |    1 |   31 |   32 ||    (14,  28) |  (28,  42) |      14 |
.  15 |    1 |   24 |   25 ||    (60,  45) |  (45,  75) |      15 |
.  16 |   21 |   18 |   39 ||    (32,  48) | (112,  64) |      16 |
.  17 |    1 |  299 |  300 ||  (2142,  17) |  (17,  34) |      17 |
.  18 |   24 |   22 |   46 ||    (54,  72) |  (90, 108) |      18 |
.  19 |    1 |  824 |  825 || (10260,  19) |  (19,  38) |      19 |
.  20 |   27 |   16 |   43 ||    (20,  40) |  (80, 100) |      20 |
.  21 |    1 |   33 |   34 ||    (42,  21) |  (21,  63) |      21 |
.  22 |    1 |  124 |  125 ||    (22,  44) |  (44,  66) |      22 |
.  23 |    1 |  945 |  946 || (12420,  23) |  (23,  46) |      23 |
.  24 |   10 |   41 |   51 ||    (96, 120) | (144, 168) |      24 |
.  25 |    1 |   26 |   27 ||    (75,  25) |  (25,  50) |      25 | .

Cf. A256918, A257120, A257475, A257218.

Haskell
```
a257478 n = a257475 n - a257120 n
```

a(37)-a(72) from Hiroaki Yamanouchi, May 03 2015

A384308 a(1) = 3; for n > 1, a(n) is the smallest number that has not appeared before and has the same set of prime divisors as a(n-1) + 1.

Original entry on oeis.org

3, 2, 9, 10, 11, 6, 7, 4, 5, 12, 13, 14, 15, 8, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 81, 82, 83, 42, 43, 44, 45, 46, 47, 36, 37, 38, 39, 40, 41, 84, 85, 86, 87, 88, 89, 60, 61, 62, 63, 32, 33, 34, 35, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 90, 91, 92, 93, 94, 95, 72, 73, 74, 75

Offset: 1

SiYang Hu, May 25 2025

nonn

Theorem 1: {a(n)} is a permutation of the positive integers greater than 1.
Proof: Suppose the smallest positive integer greater than 1 that does not appear in this sequence is m, whose set of prime divisors is {p1, p2, ..., pk}. Since there are infinitely many numbers with this set of prime divisors, we know that either all of them appear or none of them appear. Since m is the smallest among them, we know that m = p1*p2*...*pk. Therefore, p1*p2*...*pk - 1 does not appear, a contradiction, implying that all integers greater than one appear in {a(n)}. Since there are no repetitions by the definition, we have proven that {a(n)} is a permutation of the positive integers greater than 1.
From Yifan Xie, May 26 2025: (Start)
Theorem 2: The parity of a(n) is the same as the parity of n.
Proof: Since a(1) is odd, we only need to prove that an odd term is immediately followed by an even term, and an even term is immediately followed by an odd term. If a(n-1) is odd, the set of prime divisors of a(n-1) + 1 contains 2, so a(n) is even; If a(n-1) and a(n) are both even, the set of prime divisors of a(n-1) + 1 does not contain 2, so a(n)/2 is a smaller candidate for a(n), a contradiction. (End)

			For a(5) = 11, 11 + 1 = 12, its set of prime divisors is {2, 3}. The smallest number with the same set of prime divisors that has not appeared before is 6, so a(6) = 6.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Similar to A064413 and A257218.

N:= 1000:# for terms before the first term > N
for i from 2 to N do
  S:= numtheory:-factorset(i);
  if assigned(V[S]) then V[S]:= V[S] union {i}
  else V[S]:= {i}
  fi
od:
R:= 3: r:= 3: V[{3}]:= V[{3}] minus {3}:
while r < N do
  S:= numtheory:-factorset(r+1);
  if V[S] = {} then break fi;
  r:= min(V[S]);
  V[S]:= V[S] minus {r};
  R:= R, r;
od:
R; # Robert Israel, May 25 2025

s={3};Do[i=2;While[MemberQ[s,i]||First/@FactorInteger[i]!=First/@FactorInteger[s[[-1]]+1],i++];AppendTo[s,i],{n,2,81}];s (* James C. McMahon, Jun 04 2025 *)

import heapq
from math import prod
from sympy import factorint
from itertools import islice
def bgen(pset): # generator of terms with set of prime divisors = pset
    h = [prod(pset)]
    while True:
        v = heapq.heappop(h)
        yield v
        for p in pset:
            heapq.heappush(h, v*p)
def agen(): # generator of terms
    an, aset = 3, set()
    while True:
        yield an
        aset.add(an)
        an = next(m for m in bgen(set(factorint(an+1))) if m not in aset)
print(list(islice(agen(), 81))) # Michael S. Branicky, May 25 2025

cp's OEIS Frontend

A256918 Greatest common divisors of adjacent terms in A257218.

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A257122 Smallest m such that A257218(m) = n.

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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).

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A257120 Position of first occurrence of n in A256918.

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A257475 Position of second and last occurrence of n in A256918.

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A257478 Distance of positions of first and last occurrence of n in A256918.

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A384308 a(1) = 3; for n > 1, a(n) is the smallest number that has not appeared before and has the same set of prime divisors as a(n-1) + 1.

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Mathematica