A064664 -id:A064664 - cp's OEIS frontend

A064421 Term at which n appears in A064413 (if it begins at 2).

0, 1, 4, 2, 9, 3, 13, 7, 5, 8, 19, 6, 27, 12, 10, 16, 32, 11, 36, 17, 14, 18, 42, 15, 23, 26, 21, 25, 56, 22, 60, 30, 20, 31, 24, 29, 66, 35, 28, 39, 73, 34, 80, 40, 38, 41, 88, 44, 49, 45, 33, 46, 99, 47, 52, 48, 37, 55, 106, 51, 114, 59, 50, 63, 53, 58, 127, 64, 43, 54, 133

Offset: 1

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

nonn

Every nonnegative number appears here exactly once.

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. Equals A064664 - 1.

terms = 100;
ekg[s_] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++]; Append[s, k]];
EKG = Nest[ekg, {2, 4}, 2 terms];
a[1] = 0; a[n_] := FirstPosition[EKG, n] // First;
Array[a, terms] (* Jean-François Alcover, Aug 30 2018, after Robert G. Wilson v *)

More terms from Naohiro Nomoto, Sep 30 2001

A240024 Nonprime EKG sequence, cf. A064413: a(1) = 1, a(2) = 4 and for n > 2, a(n) = smallest composite number not already used which shares a factor with a(n-1).

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 9, 15, 18, 14, 16, 20, 22, 24, 21, 27, 30, 25, 35, 28, 26, 32, 34, 36, 33, 39, 42, 38, 40, 44, 46, 48, 45, 50, 52, 54, 51, 57, 60, 55, 65, 70, 49, 56, 58, 62, 64, 66, 63, 69, 72, 68, 74, 76, 78, 75, 80, 82, 84, 77, 88, 86, 90, 81, 87, 93

Offset: 1

Reinhard Zumkeller, Apr 30 2014

nonn

A239965 gives the position of the n-th nonprime; a(A239965(n))=A018252(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, EKG Sequence
Index entries for sequences related to EKG sequence

Cf. A064413, A064664 (EKG sequence).

Cf. A018252, A075570, A239965.

import Data.List (delete, genericIndex)
a240024 n = genericIndex a240024_list (n - 1)
a240024_list = 1 : ekg 4 a002808_list 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

a = {1, 4}; Do[k = 6; While[Or[PrimeQ@ k, MemberQ[a, k], CoprimeQ[a[[i - 1]], k]], k++]; AppendTo[a, k], {i, 3, 66}]; a (* Michael De Vlieger, Sep 01 2016 *)

A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

Original entry on oeis.org

1, 0, -1, 3, -7, 7, -2, -6, 9, 10, -5, -15, -14, -2, 55, 10, -17, -41, -15, -36, 42, 18, -13, 44, 81, 29, -35, -45, -18, -180, -29, -23, 41, 53, 135, 99, -48, 51, 114, 131, -30, -140, -58, -53, -303, 34, -37, -120, 34, -196, 147, -87, -45, 226, 207, 166, 103, 67, -41, 466, -84, 91, -288, 13, 350, -258, -91, -108

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Obviously, convolving this with A064664 gives A064413 back.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A064413, A064664, A323411, A349614 (Dirichlet inverse), A349615 (sum with it), A349616.

Cf. also pairs A349376, A349377 and A349397, A349398 for similar constructions.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 is prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];
A349613(n) = sumdiv(n,d,A064413(d)*A323411(n/d));

a(n) = Sum_{d|n} A064413(d) * A323411(n/d).

A255479 Inverse permutation to A255582.

Original entry on oeis.org

1, 2, 3, 4, 10, 5, 13, 6, 7, 8, 21, 9, 24, 11, 12, 16, 31, 14, 38, 18, 15, 23, 43, 20, 30, 22, 17, 25, 51, 28, 59, 27, 19, 29, 32, 37, 67, 36, 26, 34, 78, 35, 81, 39, 42, 41, 90, 44, 52, 46, 33, 48, 101, 47, 58, 50, 40, 49, 108, 55, 119, 57, 54, 64, 60, 63, 131, 66, 45, 62, 136, 68

Offset: 1

N. J. A. Sloane, Feb 27 2015

nonn

The differences |a(n)-A064664(n)| seem surprisingly small (see A255482).

About the definition: the map n -> A255582(n) is an element of the group of all permutations of the positive integers; this is the inverse of that permutation.

Cf. A098550, A255582, A064413, A064664, A255482.

Cf. A098551, A255940.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a255479 = (+ 1) . fromJust. (`elemIndex` a255582_list)
-- Reinhard Zumkeller, Mar 10 2015

A339731 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the distance between g_1 and g_n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 4, 5, 4, 5, 5, 4, 5, 6, 5, 6, 6, 5, 6, 7, 6, 7, 6, 7, 7, 8, 8, 8, 7, 6, 7, 8, 7, 8, 7, 6, 7, 8, 8, 9, 8, 8, 8, 9, 9, 9, 8, 7, 8, 9, 9, 9, 8, 7, 8, 9, 10, 10, 9, 10, 10, 10, 10, 10, 9, 8, 9, 10, 10, 10, 9, 10, 11, 11, 11, 11, 10, 10

Offset: 1

Rémy Sigrist, Dec 14 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A339731

See A339695 for a similar sequence.

Cf. A064413, A064664, A339732, A339733.

PARI
```
See Links section.
```

abs(a(n) - a(k)) <= abs(n-k) for any n, k > 0.

a(n) = A339733(n, 1).

A339732 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the number of nodes at distance n from g_1.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 11, 15, 17, 28, 39, 48, 64, 94, 116, 164, 217, 289, 395, 542, 729, 919, 1154, 1598, 2091, 2747, 3702, 4867, 6338, 8290, 10873, 14533, 18919, 24577, 31918, 41857

Offset: 0

Rémy Sigrist, Dec 14 2020

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, PARI program for A339732

Cf. A064413, A064664, A339731, A339733.

PARI
```
See Links section.
```

A339733 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; T(n, k) is the distance between g_n and g_k.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 1, 3, 3, 2, 1, 1, 2, 3, 4, 2, 2, 0, 2, 2, 4, 4, 3, 1, 1, 1, 1, 3, 4, 3, 3, 2, 1, 0, 1, 2, 3, 3, 4, 2, 2, 2, 1, 1, 2, 2, 2, 4, 5, 3, 1, 3, 2, 0, 2, 3, 1, 3, 5, 4, 4, 2, 2, 2, 1, 1, 2, 2, 2, 4, 4, 5, 3, 3, 2, 2, 2, 0, 2, 2, 2, 3, 3, 5, 5, 4, 2, 3, 1, 2, 1, 1, 2, 1, 3, 2, 4, 5

Offset: 1

Rémy Sigrist, Dec 14 2020

			Array T(n, k) begins:
  n\k|  1  2  3  4  5  6  7  8  9  10  11  12
  ---+---------------------------------------
    1|  0  1  2  2  3  3  4  4  3   4   5   4
    2|  1  0  1  1  2  2  3  3  2   3   4   3
    3|  2  1  0  1  2  1  2  2  1   2   3   2
    4|  2  1  1  0  1  1  2  3  2   2   3   3
    5|  3  2  2  1  0  1  2  2  2   1   2   3
    6|  3  2  1  1  1  0  1  2  2   2   3   3
    7|  4  3  2  2  2  1  0  1  2   2   3   2
    8|  4  3  2  3  2  2  1  0  1   1   2   1
    9|  3  2  1  2  2  2  2  1  0   1   2   1
   10|  4  3  2  2  1  2  2  1  1   0   1   2
   11|  5  4  3  3  2  3  3  2  2   1   0   1
   12|  4  3  2  3  3  3  2  1  1   2   1   0

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, Colored representation of the table for 1 <= x, y <= 1000 (where the hue is function of T(x, y))
Rémy Sigrist, PARI program for A339733

Cf. A064413, A064664, A339732, A339732.

PARI
```
See Links section.
```

T(n, n) = 0.
T(n, k) = T(k, n).
T(n, k) <= abs(n-k).
T(m, k) <= T(m, n) + T(n, k).
T(n, 1) = A339731(n).

A349616 Dirichlet convolution of A000027 (the identity function) with the Dirichlet inverse of the inverse permutation of EKG-permutation.

Original entry on oeis.org

1, 0, -2, 1, -5, 6, -7, -2, 13, 11, -9, -6, -15, 15, 49, 0, -16, -42, -18, -15, 69, 21, -20, 24, 51, 29, -48, -21, -28, -168, -30, -1, 97, 34, 150, 65, -30, 38, 141, 48, -33, -236, -38, -32, -317, 44, -42, -40, 97, -163, 163, -36, -47, 248, 192, 75, 183, 58, -48, 294, -54, 62, -443, 1, 301, -338, -61, -50, 211

Offset: 1

Antti Karttunen, Nov 23 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to EKG sequence

Cf. A000027, A064413, A064664, A323411, A349617 (Dirichlet inverse).

Cf. also A349613, A349614.

up_to = 32768;
v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with Chai Wah Wu's Dec 08 2014 Python-program given in A064413.
A064413(n) = v064413[n];
\\ Then its inverse A064664 was prepared:
m064664 = Map();
for(n=1,65539,mapput(m064664,A064413(n),n));
A064664(n) = mapget(m064664,n);
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064664(n)));
A323411(n) = v323411[n];
A349616(n) = sumdiv(n,d,d*A323411(n/d));

a(n) = Sum_{d|n} d * A323411(n/d).

A379293 Index where n appears as a term in A379248.

Original entry on oeis.org

1, 2, 9, 3, 13, 4, 42, 5, 8, 6, 88, 7, 94, 16, 11, 15, 233, 10, 241, 17, 24, 18, 412, 19, 12, 20, 23, 21, 651, 22, 659, 26, 33, 27, 44, 25, 669, 29, 35, 28, 1169, 31, 1175, 30, 32, 38, 1187, 37, 41, 14, 56, 39, 2009, 34, 46, 40, 58, 48, 2015, 49, 2021, 50, 36, 51, 96, 52, 2145, 53, 60, 54, 2151, 55, 2157, 64, 45, 63, 90, 62, 2163, 65, 57, 66, 2169, 67, 98

Offset: 1

Scott R. Shannon, Dec 20 2024

nonn

See A379248 for further details.

Scott R. Shannon, Table of n, a(n) for n = 1..1600

Cf. A379248, A379291, A379296, A064413, A064664.

A347348 a(n) is the rank of A008619(n) in A164912.

Original entry on oeis.org

1, 2, 3, 5, 4, 10, 8, 14, 6, 9, 7, 20, 13, 28, 11, 17, 12, 33, 18, 37, 15, 19, 16, 43, 24, 27, 22, 26, 23, 57, 31, 61, 21, 32, 25, 30, 36, 67, 29, 40, 35, 74, 41, 81, 39, 42, 45, 89, 46, 50, 34, 47, 48, 100, 49, 53, 38, 56, 52, 107, 60, 115, 51, 64, 54, 59

Offset: 1

Paul Curtz, Nov 21 2021

nonn

This is a permutation of the positive integers.

1, 2, 6, 9, 15, 19, ... are in a(n) and A064664(n).

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Jean-François Alcover, Plot of a(1..1000)
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A008619, A164912.

Cf. A064664.

nmax = 120;
ekg[n_] := ekg[n] = Module[{ee, k}, If[n <= 2, n, ee = Array[ekg, n - 1]; For[k = 1, True, k++, If[FreeQ[ee, k] && GCD[ekg[n - 1], k] != 1, Return[k]]]]];
b[n_] := Quotient[ekg[n] - 1, 2] + 1;
bb = Array[b, nmax];
TakeWhile[Table[Position[bb, n], {n, 1, nmax}], Length[#] == 2&] // Flatten (* Jean-François Alcover, Nov 21 2021 *)

Interleave the occurrences in A164912.

cp's OEIS Frontend

A064421 Term at which n appears in A064413 (if it begins at 2).

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Mathematica

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A240024 Nonprime EKG sequence, cf. A064413: a(1) = 1, a(2) = 4 and for n > 2, a(n) = smallest composite number not already used which shares a factor with a(n-1).

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Mathematica

A349613 Dirichlet convolution of A064413 (EKG-permutation) with the Dirichlet inverse of its inverse permutation.

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A255479 Inverse permutation to A255582.

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A339731 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the distance between g_1 and g_n.

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A339732 Let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; a(n) is the number of nodes at distance n from g_1.

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A339733 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; let G be the undirected graph with nodes {g_k, k > 0} such that for any k > 0, g_k is connected to g_{k+1} and g_{A064413(k)} is connected to g_{A064413(k+1)}; T(n, k) is the distance between g_n and g_k.

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A349616 Dirichlet convolution of A000027 (the identity function) with the Dirichlet inverse of the inverse permutation of EKG-permutation.

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A379293 Index where n appears as a term in A379248.

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