A368900 -id:A368900 - 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

A369044 LCM-transform of bijective bit reverse (A057889).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 13, 1, 11, 1, 1, 2, 17, 1, 5, 1, 1, 1, 29, 1, 19, 1, 3, 1, 23, 1, 31, 2, 1, 1, 7, 1, 41, 1, 1, 1, 37, 1, 53, 1, 1, 1, 61, 1, 1, 1, 1, 1, 43, 1, 59, 1, 1, 1, 1, 1, 47, 1, 1, 2, 1, 1, 97, 1, 3, 1, 113, 1, 73, 1, 1, 1, 89, 1, 11, 1, 1, 1, 101, 1, 1, 1, 1, 1, 1, 1, 109, 1, 1, 1, 5, 1, 67

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Bijective bit reverse, A057889, is a permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A057889(n)) = A000523(n), from which it immediately follows that A057889 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A057889(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000523, A014963, A057889, A368900, A369041, A369042, A369043, A369045.

up_to = 65537; \\ Checked up to 2^17;
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); };
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
v369044 = LCMtransform(vector(up_to,i,A057889(i)));
A369044(n) = v369044[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A057889(n)} / lcm {1..A057889(n-1)}.
a(n) = A014963(A057889(n)). [See comments.]
For n >= 1, Product_{d|n} a(A057889(d)) = n. [Implied by above.]

A369041 LCM-transform of binary Gray code (A003188).

Original entry on oeis.org

1, 3, 2, 1, 7, 5, 2, 1, 13, 1, 1, 1, 11, 3, 2, 1, 5, 3, 1, 1, 31, 29, 1, 1, 1, 23, 1, 1, 19, 17, 2, 1, 7, 1, 1, 1, 1, 53, 1, 1, 61, 1, 1, 1, 59, 1, 1, 1, 41, 43, 1, 1, 47, 1, 1, 1, 37, 1, 1, 1, 1, 1, 2, 1, 97, 1, 1, 1, 103, 101, 1, 1, 109, 1, 1, 1, 107, 1, 1, 1, 11, 1, 1, 1, 127, 5, 1, 1, 1, 1, 1, 1, 1, 113, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Binary Gray code, A003188, is a permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A003188(n)) = A000523(n), from which it immediately follows that A003188 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A003188(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16.
Index entries for sequences related to binary expansion of n

Cf. A000523, A003188, A014963, A368900, A369042, A369043, A369044.

nn = 120; a[1] = s[1] = 1; Do[s[n] = LCM[s[n - 1], BitXor[n, Floor[n/2]] ]; a[n] = s[n]/s[n - 1], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 24 2024 *)

up_to = 65537; \\ Checked up to 2^17;
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); };
A003188(n) = bitxor(n, n>>1);
v369041 = LCMtransform(vector(up_to,i,A003188(i)));
A369041(n) = v369041[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A003188(n)} / lcm {1..A003188(n-1)}.

a(n) = A014963(A003188(n)). [See comments.]

A369042 LCM-transform of the inverse of binary Gray code (A006068).

Original entry on oeis.org

1, 3, 2, 7, 1, 2, 5, 1, 1, 1, 13, 2, 3, 11, 1, 31, 1, 1, 29, 1, 5, 3, 1, 2, 17, 19, 1, 23, 1, 1, 1, 1, 1, 1, 61, 1, 1, 59, 1, 1, 7, 1, 1, 1, 1, 1, 53, 2, 1, 1, 1, 1, 1, 1, 37, 47, 1, 1, 1, 1, 41, 43, 1, 127, 1, 1, 5, 1, 11, 1, 1, 1, 113, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 103, 1, 1, 101, 1, 1, 1, 109, 1, 1, 107, 1, 2, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Inverse of Binary Gray code, A006068, is a permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A006068(n)) = A000523(n), from which it immediately follows that A006068 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A006068(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A006068, A014963, A368900, A369041, A369043, A369044.

up_to = 65537; \\ Checked up to 2^17;
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); };
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
v369042 = LCMtransform(vector(up_to,i,A006068(i)));
A369042(n) = v369042[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A006068(n)} / lcm {1..A006068(n-1)}.

a(n) = A014963(A006068(n)). [See comments.]

A369043 LCM-transform of Blue code (A193231).

Original entry on oeis.org

1, 3, 2, 5, 2, 1, 7, 1, 1, 1, 13, 1, 11, 3, 2, 17, 2, 1, 19, 1, 1, 23, 1, 1, 31, 29, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 53, 1, 1, 61, 1, 1, 1, 1, 1, 59, 1, 1, 1, 2, 1, 1, 1, 37, 1, 1, 1, 47, 1, 41, 43, 1, 1, 1, 1, 1, 1, 3, 83, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 71, 1, 1, 2, 1, 67, 1, 1, 1, 73, 1, 79, 1, 1, 1, 103, 101

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Blue code, A193231, is a self-inverse permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A193231(n)) = A000523(n), from which it immediately follows that A193231 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A193231(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A014963, A193231, A368900, A369041, A369042, A369044, A369045.

up_to = 65537; \\ Checked up to 2^17;
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); };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
v369043 = LCMtransform(vector(up_to,i,A193231(i)));
A369043(n) = v369043[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A193231(n)} / lcm {1..A193231(n-1)}.
a(n) = A014963(A193231(n)). [See comments.]
For n >= 1, Product_{d|n} a(A193231(d)) = n. [Implied by above.]

A369028 Exponential of Mangoldt function permuted by A253563.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 3, 5, 2, 1, 1, 1, 3, 1, 5, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 7, 11, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 11, 13, 2, 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, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A253563 (when viewed as an offset-1 sequence), because A253563 has the S-property explained in the comments of A368900.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A014963, A054429, A368900, A369029, A369030, A369053.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253563(n) = if(n<2,(1+n),if(!(n%2),A253560(A253563(n/2)),A253550(A253563((n-1)/2))));
A369028(n) = A014963(A253563(n));

a(n) = A014963(A253563(n)).

a(1) = 0, and for n > 0, a(n) = lcm {1..A253563(n)} / lcm {1..A253563(n-1)}. [See comments]

A369029 Exponential of Mangoldt function permuted by A253565.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A253565 (when viewed as an offset-1 sequence), because A253565 has the S-property explained in the comments of A368900.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A014963, A253565, A368900, A369028, A369030, A369053.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A253550(n) = if(1==n, 1, (n/prime(A061395(n)))*prime(1+A061395(n)));
A253560(n) = if(1==n, 1, (n*prime(A061395(n))));
A253565(n) = if(n<2,(1+n),if(!(n%2),A253550(A253565(n/2)),A253560(A253565((n-1)/2))));
A369029(n) = A014963(A253565(n));

a(n) = A014963(A253565(n)).

a(0) = 1, and for n > 0, a(n) = lcm {1..A253565(n)} / lcm {1..A253565(n-1)}. [LCM-transform, see comments]

A369030 Exponential of Mangoldt function permuted by A163511 ("Doudna-permutation mirrored").

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 1, 5, 2, 3, 1, 5, 1, 1, 1, 7, 2, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 11, 2, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 2, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A163511 (when viewed as an offset-1 sequence), because A163511 has the S-property explained in the comments of A368900, from which this can be obtained by permuting with A054429.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A014963, A054429, A163511, A365805, A368900, A369028, A369029, A369053.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A369030(n) = A014963(A163511(n));

up_to = 65537;
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); };
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
v369030 = LCMtransform(vector(up_to,i,A163511(i-1)));
A369030(n) = v369030[1+n];

a(n) = A014963(A163511(n)).
a(0) = 1, and for n > 0, a(n) = lcm {1..A163511(n)} / lcm {1..A163511(n-1)}. [See comments]
For n > 0, a(n) = A368900(1+A054429(n)).

Changed offset from 1 to 0 and swapped the main and secondary definitions. - Antti Karttunen, Jan 24 2024

A369053 Exponential of Mangoldt function permuted by A243353.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 1, 5, 5, 1, 2, 3, 1, 1, 1, 7, 7, 1, 1, 1, 3, 2, 1, 5, 1, 1, 1, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 2, 1, 5, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 12 2024

nonn

Also LCM-transform of A243353 (when viewed as an offset-1 sequence), because A243353 has the S-property explained in the comments of A368900.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A003188, A014963, A243353, A368900, A369028, A369029, A369030.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A003188(n) = bitxor(n, n>>1);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A243353(n) = A005940(1+A003188(n));
A369053(n) = A014963(A243353(n));

a(n) = A014963(A243353(n)).

a(0) = 1, and for n > 0, a(n) = lcm {1..A243353(n)} / lcm {1..A243353(n-1)}. [See comments]

A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 5, 3, 1, 2, 1, 2, 1, 1, 7, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 5, 1, 2, 1, 2, 11, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 13, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 7, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See discussion at A368900.

From the reduced formula it follows that for all i, j >= 1: A101296(i) = A101296(j) => a(i) = a(j), that is, the value of each a(n) is completely determined by its prime signature. Note that the same does not hold for related A369032.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A014963, A101296, A122111, A368900, A369032, A369033.

up_to = 2^18;
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); };
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
v369031 = LCMtransform(vector(up_to,i,A122111(i)));
A369031(n) = v369031[n];

A369031(n) = if(isprime(n),2, my(e=ispower(n,,&n)); if(e && isprime(n), prime(e), 1));

a(n) = lcm {1..A122111(n)} / lcm {1..A122111(n-1)}.
a(n) = A014963(A122111(n)). [A122111 satisfies the property S given in A368900]
If n = p^k, p prime, k >= 1, then a(n) = A000040(k), otherwise a(n) = 1.

cp's OEIS Frontend

A265576 LCM-transform of EKG sequence A064413.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A369044 LCM-transform of bijective bit reverse (A057889).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369041 LCM-transform of binary Gray code (A003188).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369042 LCM-transform of the inverse of binary Gray code (A006068).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369043 LCM-transform of Blue code (A193231).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369028 Exponential of Mangoldt function permuted by A253563.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369029 Exponential of Mangoldt function permuted by A253565.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369030 Exponential of Mangoldt function permuted by A163511 ("Doudna-permutation mirrored").

Views

Author

Keywords

Comments