A300012 -id:A300012 - cp's OEIS frontend

A302781 Divisor-or-multiple permutation of natural numbers constructed from two-dimensional Hilbert curve (A163357) and Fermi-Dirac primes (A050376).

1, 2, 6, 3, 15, 5, 10, 30, 120, 40, 20, 60, 12, 24, 8, 4, 28, 84, 168, 56, 14, 7, 21, 42, 210, 105, 35, 70, 280, 840, 420, 140, 1260, 3780, 7560, 2520, 630, 315, 945, 1890, 378, 189, 63, 126, 504, 1512, 756, 252, 36, 72, 216, 108, 540, 180, 360, 1080, 270, 90, 45, 135, 27, 54, 18, 9, 117, 351, 702, 234, 936, 468

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Note that the starting offset is 0, to align with A052330 and A207901.
Shares with A064736, A207901, A298480, A302350, A302783, A303771, etc. the property that a(n) is either a divisor or a multiple of a(n+1). Permutations satisfying such property are called "divisor-or-multiple permutations" in the OEIS, although Mazet & Saias call them "chain permutations" in their paper. [Edited by Antti Karttunen, Aug 26 2018]
One way to construct such permutations is by composing A052330 from the right with any such permutation like A003188 or A302846 where the binary expansions of a(n) and a(n+1) always differ by just a single bit-position.
Further permutations satisfying the same condition could be constructed from higher-dimensional versions (i.e., greater than 2) of Hilbert's space-filling curves, where the coordinates of each dimension would be Gray coded separately and then interleaved together. Permutation A207901 is essentially a one-dimensional variant of the same idea, while this is constructed from the 2-dimensional curve A163357, which is a Hamiltonian path on N X N grid.
See Peter Munn's A300012 for another idea for constructing such a permutation. - Antti Karttunen, Aug 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..16383
Pierre Mazet and Eric Saias, Etude du graphe divisoriel 4, arXiv:1803.10073 [math.NT], 2018.
Various, Discussion on SeqFan-list, April 2018.
Index entries for sequences that are permutations of the natural numbers

Cf. A302782 (inverse).

Cf. A003188, A050376, A052330, A059252, A059253, A064706, A163356, A163357, A207901, A298480, A300012, A302844, A302846, A302783, A303771.

up_to_e = 14;
v050376 = vector(up_to_e);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A064706(n) = bitxor(n, n>>2);
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r))));
A302781(n) = A052330(A064706(A163356(n)));

a(n) = A052330(A302846(n)), where A302846(n) = A000695(A003188(A059253(n))) + 2*A000695(A003188(A059252(n))).

a(n) = A207901(A302844(n)) = A052330(A064706(A163356(n))).

Name edited by Antti Karttunen, Aug 26 2018

A340114 Table T(n,k), n>=1, k>=1, row n being the lexicographically earliest of the longest sequences of distinct positive integers in which the k-th term does not exceed n*k and the smaller of adjacent terms divides the larger, giving a prime.

Original entry on oeis.org

1, 2, 1, 3, 6, 2, 4, 8, 1, 2, 4, 12, 6, 18, 9, 3, 15, 30, 10, 5, 35, 7, 21, 42, 14, 28, 56, 8, 16, 48, 24, 72, 36, 2, 1, 3, 9, 18, 6, 12, 24, 8, 40, 20, 10, 5, 55, 11, 33, 66, 22, 44, 4, 52, 26, 78, 39, 13, 91, 7, 49, 98, 14, 42, 126, 63, 21, 105, 15, 45, 135, 27, 81

Offset: 1

Peter Munn, Dec 28 2020

The longest sequence is finite for all n. We can deduce this, because we know from the work of Saias that A337125(m)/m * log m is bounded, where A337125(m) is the length of the longest simple path in the divisor graph of {1,...,m}. See the comment in A337125 giving constraints on its terms.

The sequence of row lengths starts 2, 6, 25, 97.

			For n = 1, the only sequences of distinct positive integers that have their k-th term not exceeding 1*k = k, are those whose n-th term is k. The longest such sequence in which the smaller of adjacent terms divides the larger, giving a prime, is (1, 2), since 3/2 is 1.5. So row 1 has length 2, with T(1,1) = 1, T(1,2) = 2.
Table begins:
1, 2;
1, 3, 6, 2, 4, 8;
1, 2, 4, 12, 6, 18, 9, 3, 15, 30, 10, 5, 35, 7, 21, 42, 14, 28, 56, 8, 16, 48, 24, 72, 36;
...

Peter Munn, Rows n = 1..4 of triangle, flattened
E. Saias, Applications des entiers à diviseurs denses, Acta Arithmetica, 83, 3 (1998), 225-240.

Cf. A000040, A300012, A337125, A339491.

For n >= 1, 1 <= k <= row length(n), T(n,k) <= n * k.

For n >= 1, 1 <= k < row length(n), max(T(n,k+1)/T(n,k), T(n,k)/T(n,k+1)) is in A000040.

cp's OEIS Frontend

A302781 Divisor-or-multiple permutation of natural numbers constructed from two-dimensional Hilbert curve (A163357) and Fermi-Dirac primes (A050376).

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