A152458 -id:A152458 - 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

A353989 a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has a 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 3, 6, 2, 10, 8, 12, 4, 14, 7, 21, 9, 15, 5, 20, 16, 18, 22, 11, 33, 27, 24, 26, 13, 39, 30, 25, 35, 40, 28, 36, 32, 34, 38, 19, 57, 42, 44, 46, 23, 69, 45, 48, 50, 52, 54, 51, 17, 85, 55, 60, 56, 49, 63, 66, 58, 29, 87, 72, 62, 31, 93, 75, 65, 70, 64, 68, 74, 76, 78, 80, 82, 84, 77, 88, 86, 43

Offset: 1

Scott R. Shannon, May 13 2022

nonn

This sequence is similar to the EKG sequence A064413 with the additional restriction that each term must share at least one 1-bit in common with the previous term in their binary expansions. The majority of terms are concentrated along the same three lines as in A064413 although at least three additional lines appear that contains fewer terms. See the linked image. Unlike A064413 the primes do not occur in their natural order and a prime p can be preceded and followed by multiples of p other than 2p and 3p respectively.
In the first 100000 terms the fixed points are 1, 16, 32, 209, 527, and it is likely no more exist. In the same range the lowest unseen number is 34849; the sequence is conjectured to be a permutation of the positive integers.
See A353245 for the binary AND operation of each pair of terms.

			a(3) = 6 as a(2) = 3, 6 = 110_2, 3 = 11_2, and 6 is the smallest unused number that shares a common factor with 3 and has a 1-bit in common with 3 in their binary expansions.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A064413, A353245, A152458, A353712, A352633.

A352763 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 6, 9, 18, 12, 3, 24, 32, 10, 5, 40, 16, 14, 48, 15, 80, 34, 17, 68, 26, 36, 27, 96, 20, 35, 28, 64, 22, 33, 30, 65, 50, 72, 21, 42, 69, 138, 52, 66, 44, 82, 41, 656, 38, 88, 128, 46, 144, 39, 192, 45, 130, 13, 208, 256, 54, 129, 60, 194, 56, 7, 112, 132, 11, 176, 70, 25, 100, 136, 51

Offset: 1

Scott R. Shannon, May 15 2022

nonn

This sequence is similar to the EKG sequence A064413 with the additional restriction that no term can have a 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being much larger than their preceding term, while the smaller terms overall show similar behavior to A109812. See the linked image. Unlike A064413 the primes do not occur in their natural order and the term following a prime can be a very large multiple of the prime.

In the first 50000 terms the fixed points are 1, 2, 105, 135, 225, 2157, 3972, 7009, 8531, although it is likely more exist. In the same range the lowest unseen number is 383; the sequence is conjectured to be a permutation of the positive integers.

			a(5) = 6 as a(4) = 8, 6 = 110_2, 8 = 1000_2, and 6 is the smallest unused number that shares a common factor with 8 but has no 1-bit in common with 8 in their binary expansions.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms for values below 15000. The green line is y = n. The maximum value in this range is a(8045) = 959024.

Cf. A353989, A109812, A064413, A152458, A353712, A352633.

A354087 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and whose binary expansion has a single 1-bit in common with the binary expansion of a(n-1).

Original entry on oeis.org

1, 3, 6, 2, 10, 8, 12, 4, 14, 18, 15, 20, 5, 25, 35, 21, 9, 24, 16, 22, 11, 33, 27, 48, 26, 13, 52, 32, 34, 30, 36, 28, 7, 42, 49, 77, 56, 38, 19, 133, 57, 69, 46, 66, 39, 65, 45, 50, 40, 54, 68, 44, 70, 58, 72, 60, 74, 64, 76, 80, 55, 88, 96, 51, 78, 81, 102, 130, 62, 132, 63, 129, 43, 86, 104, 82

Offset: 1

Scott R. Shannon, May 17 2022

This sequence is similar to the EKG sequence A064413 with the additional restriction that each term must share a single 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being significantly larger than their preceding term, while the smaller terms overall show similar behavior to A109812. See the linked image. Unlike A064413 the primes do not occur in their natural order and both the proceeding and following terms of the primes can be large multiples of the prime.

In the first 100000 terms the fixed points are 1, 3, 30, 38, 350, 1603, 1936, 10176, 11976, 46123, 58471, 89870, although it is likely more exist. In the same range the lowest unseen number is 1019; the sequence is conjectured to be a permutation of the positive integers.

			a(6) = 8 as a(5) = 10, 8 = 1000_2, 10 = 1010_2, and 8 is the smallest unused number that shares a common factor with 10 and has a single 1-bit in common with 10 in their binary expansions. Note that 4 satisfies the first criterion but not the second.

Scott R. Shannon, Image of the first 100000 terms for values less than 200000. The green line is y = n.

Cf. A353989, A064413, A353245, A152458, A353712, A352633.

A382702 Indices k where b(k) > k, where b is the EKG sequence A064413.

Original entry on oeis.org

3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 108

Offset: 1

N. J. A. Sloane, Apr 04 2025

nonn

			A064413(3) = 4 > 3, so 3 is a term.

N. J. A. Sloane, Table of n, a(n) for n = 1..9296

Cf. A064413, A152458, A382703-A382708.

A382708 Number of triples (i,j,k), 1 <= i < j < k <= n such that A064413(i) < A064413(k) < A064413(j).

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 4, 14, 20, 39, 39, 39, 59, 97, 97, 97, 134, 162, 177, 260, 260, 280, 300, 360, 360, 423, 525, 694, 694, 722, 817, 895, 1129, 1129, 1162, 1254, 1546, 1546, 1615, 1751, 1856, 1925, 2326, 2326, 2436, 2546, 2625, 2704, 2783, 3061, 3104, 3196, 3415, 3458, 3458, 3699, 4439, 4439, 4590, 4890, 5725, 5725, 5842

Offset: 1

N. J. A. Sloane, Apr 04 2025

nonn

This is the number of occurrences of the pattern "132" in the first n terms of A064413.

			The first 5 terms of A064413 are 1, 2, 4, 6, 3, and at that point we can see four occurrences of the pattern "132", namely the triples 143, 163, 243, and 263, so a(5) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..100

Cf. A064413, A152458, A382702-A382707.

A247379 a(n) = gcd(EKG(n),n) with EKG = A064413.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 8, 1, 5, 1, 6, 1, 7, 3, 8, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 6, 1, 2, 1, 17, 7, 2, 1, 19, 3, 40, 1, 2, 1, 1, 3, 2, 1, 6, 7, 1, 3, 4, 1, 1, 5, 2, 1, 29, 1, 2, 1, 31, 9, 64, 1, 2, 1, 1, 3, 2, 1, 8, 1, 1, 3, 1, 7, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Sep 15 2014

nonn

a(n) = gcd(A064413(n),n);

a(A247383(n)) = n and a(m) != n for m < A247383(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A064413, A152458 (fixed points), A247383.

Haskell
```
a247379 n = gcd n $ a064413 n
```

A065519 a(n) = A064413(n)-n.

Original entry on oeis.org

0, 0, 1, 2, -2, 3, 5, 0, 1, -5, 4, 6, 1, -7, 6, 8, -1, 2, 3, -9, 12, 5, 7, 1, 10, 2, -1, -15, 10, 6, 1, 2, -16, 17, 7, 2, -18, 19, 6, 0, 3, 4, -20, 25, 3, 4, 5, 6, 7, -1, 12, 8, 2, 11, 15, 2, -28, 29, 7, 2, -30, 31, 9, 0, 3, 8, -30, 43, 6, 8, 5, 8, 9

Offset: 1

N. J. A. Sloane, Nov 28 2001

sign

Zeros appear at A152458.

Peter Kagey, Table of n, a(n) for n = 1..10000
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.
Index entries for sequences related to EKG sequence

Cf. A064413, A152458.

terms = 100;
ekg[s_] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++]; Append[s, k]];
Nest[ekg, {1, 2}, terms-2] - Range[terms] (* Jean-François Alcover, Sep 10 2018, after Robert G. Wilson v *)

A379292 Number k such that A379248(k) = k.

Original entry on oeis.org

1, 2, 33, 155, 913, 1145

Offset: 1

Scott R. Shannon, Dec 20 2024

These are the fixed points in the first 500000 terms of A379248. See that sequence for further details.

Cf. A379248, A379291, A379296, A064413, A152458.

A382703 Numbers k which appear "prematurely" in A064413, that is, k appears before the k-th term.

Original entry on oeis.org

4, 6, 9, 12, 10, 15, 18, 14, 21, 24, 20, 22, 33, 27, 30, 25, 35, 28, 39, 36, 32, 34, 51, 42, 38, 57, 45, 44, 46, 69, 48, 50, 52, 54, 56, 63, 60, 55, 65, 70, 58, 87, 66, 62, 93, 72, 68, 74, 111, 75, 78, 76, 80, 82, 123, 81, 84, 88, 86, 129, 90, 85, 95, 100, 92, 94, 141, 96, 98, 104, 102, 99, 105, 108, 106, 159, 114, 110, 112, 116

Offset: 1

N. J. A. Sloane, Apr 04 2025

nonn

a(n) = A064413(A382702(n)).

			4 appears in A064413 at the third term, so 4 is a term.

N. J. A. Sloane, Table of n, a(n) for n = 1..9296

Cf. A064413, A152458, A382702-A382708.

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).

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A353989 a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has a 1-bit in common with the binary expansion of a(n-1).

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A352763 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

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A354087 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and whose binary expansion has a single 1-bit in common with the binary expansion of a(n-1).

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A382702 Indices k where b(k) > k, where b is the EKG sequence A064413.

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A382708 Number of triples (i,j,k), 1 <= i < j < k <= n such that A064413(i) < A064413(k) < A064413(j).

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A247379 a(n) = gcd(EKG(n),n) with EKG = A064413.

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A065519 a(n) = A064413(n)-n.

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Mathematica

A379292 Number k such that A379248(k) = k.

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