cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A265576 LCM-transform of EKG sequence A064413.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 02 2016

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

More terms from Antti Karttunen, Apr 21 2025

A265575 LCM-transform of Euler totient numbers (A000010).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 11, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 2, 1, 13, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 41, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A000010.
Other LCM-transforms are A061446, A265574, A265576, A265577, A265578.

Programs

  • Maple
    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;
with(numtheory);
t1:=[seq(phi(n),n=1..100)];
LCMXfm(t1);
  • Mathematica
    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]];
LCMXfm[Table[EulerPhi[n], {n, 1, 100}]] (* Jean-François Alcover, Dec 05 2017, from Maple *)
  • PARI
    up_to = 10000;
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); };
v265575 = LCMtransform(vector(up_to,i,eulerphi(i)));
A265575(n) = v265575[n]; \\ Antti Karttunen, Nov 09 2018

A265577 LCM-transform of Yellowstone permutation A098550.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 5, 7, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 3, 11, 13, 1, 1, 1, 1, 1, 1, 17, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 19, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 3, 1, 29, 31, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A064413.
Other LCM-transforms are A061446, A265574, A265575, A265576.

Programs

  • Maple
    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 A098550
LCMXfm(t1);
  • Mathematica
    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]];
y[n_ /; n <= 3] := n; y[n_] := y[n] = For[k = 1, True, k++, If[ FreeQ[ Array[y, n-1], k], If[GCD[k, y[n-1]] == 1 && GCD[k, y[n-2]] > 1, Return[k]]]];
Yperm = Array[y, 100];
LCMXfm[Yperm] (* Jean-François Alcover, Dec 03 2017 *)

A265578 LCM-transform of number of divisors function (A000005).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2016

Keywords

Comments

Terms larger than one occur at n = 2, 4, 6, 16, 24, 36, 64, 120, 840, 900, 1296, 7560, 44100, 46656, 83160, ... - Antti Karttunen, Nov 06 2018

Links

Crossrefs

Cf. A000005.
Other LCM-transforms are A061446, A265574, A265575, A265576, A265577.

Programs

  • Maple
    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;
with(numtheory);
t1:=[seq(tau(n),n=1..100)];
LCMXfm(t1);
  • Mathematica
    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]];
LCMXfm[Table[DivisorSigma[0, n], {n, 1, 100}]] (* Jean-François Alcover, Dec 05 2017, from Maple *)
  • PARI
    up_to = 16384;
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); };
v265578 = LCMtransform(vector(up_to,i,numdiv(i)));
A265578(n) = v265578[n]; \\ Antti Karttunen, Nov 06 2018
Showing 1-4 of 4 results.

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