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 1-9 of 9 results.

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

Views

Author

Antti Karttunen, Apr 22 2025

Keywords

Links

Crossrefs

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).

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

Views

Author

Antti Karttunen, Apr 22 2025

Keywords

Comments

Restricted growth sequence transform of A265576.

Links

Crossrefs

Cf. A000720, A064413, A265576, A383285 (positions of terms > 1).
Positions of records: {2} U A064423.

Programs

  • PARI
    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];

Formula

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

A368900 LCM-transform of Doudna sequence.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 3, 2, 7, 1, 1, 1, 5, 1, 3, 2, 11, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 2, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 2, 17, 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, 13, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

David James Sycamore and Antti Karttunen, Jan 10 2024

Keywords

Comments

Let's define "property S" for sequences as follows: If s is any sequence of positive natural numbers, normalized to begin with offset 1, then it satisfies the S-property if LCM-transform(s) is equal to the sequence obtained by applying A014963 to sequence s, or in other words, when for all n >= 1, lcm {s(1)..s(n)} / lcm {s(1)..s(n-1)} = A014963(s(n)). This holds if and only if, for all n >= 1, when, either (case A): s(n) is of the form p^k, p prime, then gcd(s(n), lcm {s(1)..s(n-1)}) must be equal to p^(k-1), or (case B): when s(n) is not a prime power, then gcd(s(n), lcm {s(1)..s(n-1)}) must be equal to s(n). Together the cases (A) and (B) reduce to the condition that each prime power should appear in s before any of its multiples do.
Clearly the Doudna-sequence satisfies the property by the way of its construction, as do many of its variants like A356867 (see A369060).
Also, for any base-2 related permutation b that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., if for all n >= 1, A000523(b(n)) = A000523(n), then the above property is automatically satisfied.
Furthermore, because in Doudna-sequence no multiple of any term is located on the same row as the term itself (see the tree-illustration in A005940), it follows that any composition of A005940 with any such base-2 related permutation as mentioned above also automatically satisfies the S-property, for example, the permutations A163511, A243353, A253563, A253565, A366260, A366263 and A366275.
Note: Like A005940 itself, also this sequence might be more logical with the starting offset 0 instead of 1, to better align with the underlying mapping from the binary expansion of n to the prime factorization. - Antti Karttunen, Jan 24 2024

Links

Crossrefs

Cf. A000040, A000225, A000265, A005940, A005941, A007814, A023758, A368901.
List of LCM-transforms of permutations (permutation given in parentheses):
Cf. A265576 (A064413; note that the EKG sequence permutation does not satisfy the S-property).
In all following cases, the permutation satisfies the S-property:
Cf. A014963 (A000027),
Cf. A369028 (A253563), A369029 (A253565), A369030 (A163511), A369053 (A243353).
Cf. A369031 (A122111), A369032 (A241909), A369033 (A241916).
Cf. A369041 (A003188), A369042 (A006068), A369043 (A193231), A369044 (A057889), A369041 (A054429). [Base-2 related permutations]
Cf. A369060 (A356867).
Other permutations that have the same property: A303767, (and when used as an offset=1 sequence): A052330.

Programs

  • Mathematica
    nn = 120; Array[Set[{s[#], a[#]}, {#, #}] &, 2]; j = 2;
Do[If[EvenQ[n],
  Set[s[n], 2 s[n/2]],
  Set[s[n],
    Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &,
      FactorInteger[s[(n + 1)/2]]]]];
  k = LCM[j, s[n]]; a[n] = k/j; j = k, {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 24 2024 *)
  • 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); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t) };
v368900 = LCMtransform(vector(up_to,i,A005940(i)));
A368900(n) = v368900[n];
  • PARI
    A000265(n) = (n>>valuation(n,2));
A209229(n) = (n && !bitand(n,n-1));
A368900(n)  = if(1==n, 1, my(x=A000265(n-1)); if(A209229(1+x), prime(1+valuation(n-1,2)), 1));

Formula

a(n) = A368901(n) / A368901(n-1) = lcm {1..A005940(n)} / lcm {1..A005940(n-1)}.
a(n) = A005940(n) / gcd(A005940(n), A368901(n-1)).
a(n) = A014963(A005940(n)). [Because A005940 satisfies the property given in the comments]
For n >= 1, Product_{d|n} a(A005941(d)) = n. [Implied by above]
For n >= 1, a(n) = A369030(1+A054429(n-1)).
For n > 1, if n-1 is a number of the form 2^i - 2^j with i >= j, then a(n) = prime(1+j), otherwise a(n) = 1.

A383295 Positions of proper prime powers (A246547) in EKG-sequence.

Original entry on oeis.org

3, 6, 8, 17, 22, 24, 31, 50, 64, 76, 112, 122, 124, 171, 232, 240, 290, 319, 359, 485, 521, 595, 696, 823, 947, 982, 1279, 1313, 1642, 1810, 1961, 2090, 2096, 2168, 2306, 2736, 3002, 3398, 3638, 3932, 4379, 4733, 4913, 5207, 6072, 6312, 6583, 6710, 7717, 7898, 9165, 9929, 10298, 11144, 11568, 11786, 12430, 14138
Offset: 1

Views

Author

Antti Karttunen, Apr 22 2025

Keywords

Comments

Apparently also numbers k for which A265576(k) > 1 and A064413(k) is neither 2 nor 2*odd prime.

Links

Crossrefs

Cf. A064413, A246547, A265576.
Setwise difference A383294 \ A064955.
Conjectured to be a subsequence of A383285.

Programs

  • PARI
    isA383295(n) = { my(x=A064413(n)); (isprimepower(x) && !isprime(x)); };

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

A383293 Exponential of Mangoldt function applied to EKG-sequence: a(n) = A014963(A064413(n)).

Original entry on oeis.org

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

Antti Karttunen, Apr 22 2025

Keywords

Links

Crossrefs

Cf. A014963, A064413, A383294 (positions of terms > 1).
Cf. also A265576.

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

A383258 LCM-transform of A064664 (the inverse of the EKG-sequence).

Original entry on oeis.org

1, 2, 5, 3, 1, 2, 7, 2, 1, 3, 1, 1, 1, 13, 11, 17, 1, 1, 37, 1, 1, 19, 43, 2, 1, 3, 1, 1, 1, 23, 61, 31, 1, 2, 5, 1, 67, 1, 29, 1, 1, 1, 3, 41, 1, 1, 89, 1, 1, 1, 1, 47, 1, 1, 53, 7, 1, 1, 107, 1, 1, 1, 1, 2, 1, 59, 2, 1, 1, 1, 1, 1, 1, 1, 1, 71, 1, 1, 151, 1, 1, 73, 1, 1, 1, 1, 1, 79, 167, 83, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 197
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2025

Keywords

Comments

As the sequence A064664 has no S-property defined in the comments of A368900, therefore this is not equal to A014963(A064664(n)).

Links

Crossrefs

Cf. A064413, A064664, A064954.
Cf. also A265576, A368900.

Formula

a(n) = lcm {1..A064664(n)} / lcm {1..A064664(n-1)}.
Showing 1-9 of 9 results.

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