A369029 -id:A369029 - cp's OEIS frontend

A368900 LCM-transform of Doudna sequence.

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

David James Sycamore and Antti Karttunen, Jan 10 2024

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..65537
A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966. [Defines the LCM-transform operation]
Index entries for sequences related to binary expansion of n

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.

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 *)

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];

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));

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.

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]

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]

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A368900 LCM-transform of Doudna sequence.

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A369028 Exponential of Mangoldt function permuted by A253563.

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

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A369053 Exponential of Mangoldt function permuted by A243353.

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