A064423 -id:A064423 - cp's OEIS frontend

A265576 LCM-transform of EKG sequence A064413.

1, 2, 2, 3, 1, 3, 1, 2, 5, 1, 1, 1, 7, 1, 1, 1, 2, 1, 11, 1, 1, 3, 1, 5, 1, 1, 13, 1, 1, 1, 2, 17, 1, 1, 1, 19, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 29, 1, 1, 1, 31, 1, 1, 1, 2, 1, 37, 1, 1, 1, 1, 1, 1, 41, 1, 1, 3, 1, 1, 1, 43, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 53

Offset: 1

N. J. A. Sloane, Jan 02 2016

nonn

This is not equal to A383293(n) = A014963(A064413(n)) because the EKG-permutation doesn't satisfy the property that all prime powers should appear before any of their multiples, as, for example, A064413(4) = 6 comes before A064413(5) = 3. See comments in A368900. - Antti Karttunen, Jan 13 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Andrzej Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
Andrzej Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
Index entries for sequences related to EKG sequence.

Cf. A064413, A383284 (rgs-transform), A383285 (positions of terms > 1), A383295.
Positions of records: {2} U A064423.
Other LCM-transforms are A014963, A061446, A265574, A265575, A368900 (see the last one for many other examples), A383258.
Cf. also A383293.

LCMXfm:=proc(a) local L,i,n,g,b;
L:=nops(a);
g:=Array(1..L,0); b:=Array(1..L,0);
b[1]:=a[1]; g[1]:=a[1];
for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
lprint([seq(b[i],i=1..L)]);
end;
# let t1 contain the first 100 terms of A064413
LCMXfm(t1);

LCMXfm[a_List] := Module[{L = Length[a], b, g}, b[1] = g[1] = a[[1]]; b[] = 0; g[] = 0; Do[g[n] = LCM[g[n - 1], a[[n]]]; b[n] = g[n]/g[n - 1], {n, 2, L}]; Array[b, L]];
ekg[1] = 1; ekg[2] = 2; ekg[n_] := ekg[n] = For[k = 1, True, k++, If[FreeQ[ Array[ekg, n - 1], k] && !CoprimeQ[k, ekg[n - 1]], Return[k]]];
LCMXfm[Array[ekg, 100]] (* Jean-François Alcover, Dec 05 2017 *)

LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2, len, g[n] = lcm(g[n-1], v[n]); b[n] = g[n]/g[n-1]); (b); };
up_to = 20000;
v265576 = LCMtransform(vector(up_to, i, A064413(i))); \\ With precomputed A064413.
A265576(n) = v265576[n]; \\ Antti Karttunen, Apr 21 2025

a(n) = lcm {1..A064413(n)} / lcm {1..A064413(n-1)}. - Antti Karttunen, Apr 21 2025

More terms from Antti Karttunen, Apr 21 2025

A304526 Möbius transform of A064664, the inverse of EKG-sequence.

Original entry on oeis.org

1, 1, 4, 1, 9, -2, 13, 5, 1, -2, 19, 2, 27, -2, -3, 9, 32, 7, 36, 8, -3, -2, 42, 4, 14, -2, 16, 12, 56, 15, 60, 14, -3, -2, 2, 15, 66, -2, -3, 17, 73, 23, 80, 21, 27, -2, 88, 20, 36, 23, -3, 19, 99, 20, 24, 18, -3, -2, 106, 18, 114, -2, 35, 33, 17, 41, 127, 32, -3, 33, 133, 24, 137, -2, 44, 34, 45, 44, 150, 23, 54, -2, 159, 27, 42, -2, -3, 33, 166, 25, 52

Offset: 1

Antti Karttunen, May 18 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A000010, A008683, A064423, A064664, A304527, A327867 (even bisection), A323411, A349617.

A304526(n) = sumdiv(n, d, moebius(n/d)*A064664(d));

a(n) = Sum_{d|n} A008683(n/d)*A064664(d).
a(n) = A064664(n) - A304527(n).
For all n >= 1, a(A000040(n)) = A064423(n).
For n >= 2, a(2*A000040(n)) = -2.
For n >= 3, a(3*A000040(n)) = -3.
a(n) = Sum_{d|n} A000010(n/d)*A349617(d). - Antti Karttunen, Jan 27 2024

A064425 Gaps between where primes occur in A064413.

Original entry on oeis.org

3, 5, 4, 6, 8, 5, 4, 6, 14, 4, 6, 7, 7, 8, 11, 7, 8, 13, 6, 4, 13, 9, 7, 15, 7, 8, 6, 4, 9, 20, 17, 6, 5, 15, 8, 10, 8, 9, 14, 8, 8, 19, 8, 6, 4, 20, 19, 10, 8, 6, 14, 4, 12, 13, 14, 9, 7, 13, 10, 9, 8, 24, 6, 12, 7, 23, 10, 16, 9, 6, 11, 18, 8, 17, 6, 6, 20, 10, 15

Offset: 1

N. J. A. Sloane, Oct 01 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
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. First differences of A064955 and also of A064423.

Cf. A137847.

terms = 80;
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}, 12 terms];
fp[n_] := FirstPosition[EKG, Prime[n]][[1]];
Array[fp, terms] // Differences (* Jean-François Alcover, Sep 02 2018, after Robert G. Wilson v in A064413 *)

A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

Original entry on oeis.org

1, 5, 6, 11, 10, 24, 39, 22, 15, 14, 25, 69, 21, 20, 53, 130, 76, 51, 29, 28, 54, 112, 97, 50, 78, 96, 34, 33, 84, 209, 232, 38, 37, 85, 153, 44, 43, 111, 156, 179, 109, 58, 57, 142, 383, 140, 148, 352, 281, 124, 249, 299, 93, 118, 218, 62, 61, 143, 172, 68, 67, 173, 641, 696, 162, 75, 74, 210, 227, 238, 191, 535, 82, 81

Offset: 1

Antti Karttunen, May 31 2018

nonn

Permutation of A064957.

Cf. A305294 (a left inverse).

Cf. A003961, A064413, A064423, A064664, A064955, A064957.

a(n) = A064664(A003961(A064413(n))).
Other identities. For all n >= 1:
A305294(a(n)) = n.
a(A064955(n)) = A064955(1+n).
For all n >= 2, a(A064423(n)) = 1+A064955(1+n).

A383285 Positions of terms > 1 in A265576, where A265576 is the LCM-transform of EKG-sequence.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 13, 17, 19, 22, 24, 27, 31, 32, 36, 42, 50, 56, 60, 64, 66, 73, 76, 80, 88, 99, 106, 112, 114, 122, 124, 127, 133, 137, 150, 159, 166, 171, 181, 188, 196, 202, 206, 215, 232, 235, 240, 252, 258, 263, 278, 286, 290, 296, 304, 313, 319, 327, 335, 343, 359, 362, 370, 376, 380, 400, 419, 429, 437, 443

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..22747
Index entries for sequences related to EKG sequence.

Positions of terms larger than one in A265576 and in A383284.
Cf. A064413.
Cf. A064423 (after its initial 1 is a subsequence of this sequence), A383295 (conjectured subsequence).

A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 1, 5, 5, 4, 3, 10, 10, 9, 2, 1, 5, 14, 14, 13, 8, 4, 6, 11, 10, 20, 20, 19, 3, 1, 28, 28, 27, 9, 33, 33, 32, 7, 5, 14, 37, 37, 36, 2, 6, 20, 8, 10, 24, 18, 4, 15, 21, 11, 43, 43, 42, 13, 57, 57, 56, 3, 1, 28, 61, 61, 60, 12, 19, 33, 5, 67, 67, 66, 17, 9, 25, 14, 74, 74, 73, 7, 29, 34, 6, 37, 81, 81, 80, 2

Offset: 1

Antti Karttunen, May 31 2018

nonn

Cf. A064413, A064423, A064955, A064664, A064989, A305293.

a(n) = A064664(A064989(A064413(n))).
For all n >= 1, a(A305293(n)) = n.
For all n >= 2, a(A064423(n)) = a(A064955(n)) = A064955(n-1).
For all n >= 3, a(1+A064955(n)) = A064423(n-1).

A383284 Lexicographically earliest infinite sequence such that a(i) = a(j) => A265576(i) = A265576(j), for all i, j >= 1, where A265576 is the LCM-transform of EKG-sequence.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 1, 2, 4, 1, 1, 1, 5, 1, 1, 1, 2, 1, 6, 1, 1, 3, 1, 4, 1, 1, 7, 1, 1, 1, 2, 8, 1, 1, 1, 9, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 12, 1, 1, 1, 2, 1, 13, 1, 1, 1, 1, 1, 1, 14, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Restricted growth sequence transform of A265576.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to EKG sequence.

Cf. A000720, A064413, A265576, A383285 (positions of terms > 1).

Positions of records: {2} U A064423.

up_to = 100000;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2, len, g[n] = lcm(g[n-1], v[n]); b[n] = g[n]/g[n-1]); (b); };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v064413 = readvec("b064413_huge_to.txt"); \\ From b-file of A064413 computed previously.
A064413(n) = v064413[n];
v383284 = rgs_transform(LCMtransform(vector(up_to, i, A064413(i))));
A383284(n) = v383284[n];

a(n) = 1+A000720(A265576(n)). [Conjectured. True if there are no composites in A265576]

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A265576 LCM-transform of EKG sequence A064413.

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A304526 Möbius transform of A064664, the inverse of EKG-sequence.

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A064425 Gaps between where primes occur in A064413.

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A305293 Prime shift towards larger primes, conjugated by the EKG-permutation: a(n) = A064664(A003961(A064413(n))).

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A383285 Positions of terms > 1 in A265576, where A265576 is the LCM-transform of EKG-sequence.

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A305294 Prime shift towards smaller primes, conjugated by the EKG-permutation: a(n) = A064664(A064989(A064413(n))).

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A383284 Lexicographically earliest infinite sequence such that a(i) = a(j) => A265576(i) = A265576(j), for all i, j >= 1, where A265576 is the LCM-transform of EKG-sequence.

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