A304755 -id:A304755 - cp's OEIS frontend

A304757 Restricted growth sequence transform of A046523(A304755(n)).

1, 2, 3, 2, 4, 5, 6, 4, 5, 7, 8, 9, 10, 8, 9, 7, 11, 3, 2, 4, 12, 13, 3, 14, 8, 11, 12, 4, 15, 16, 3, 2, 4, 12, 13, 3, 14, 8, 11, 12, 4, 15, 17, 3, 13, 18, 13, 19, 12, 20, 8, 14, 21, 14, 8, 22, 23, 12, 19, 24, 3, 2, 4, 12, 13, 3, 14, 8, 11, 12, 4, 15, 16, 13, 3, 12, 19, 18, 13, 21, 14, 20, 19, 12, 25, 26, 27, 28, 29, 30, 26, 31, 32, 33, 4, 5, 6, 15, 12, 4

Offset: 1

Antti Karttunen, May 21 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A304755.

Compare also to the scatter plots of A304098, A304729, A304730 and A304732.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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; };
v304757 = rgs_transform(vector(up_to,n,A046523(A304755(n)))); \\ Needs also code from A304755
A304757(n) = v304757[n];

A304756 Inverse of A304755: if A304755(k) = n, a(n) = k, or 0 if n does not occur in A304755.

Original entry on oeis.org

1, 2, 4, 6, 19, 3, 32, 12, 9, 18, 62, 5, 488, 31, 23, 87, 1449, 8, 5505, 20, 36, 61, 14658, 11, 96, 487, 15, 33, 50871, 22, 190668, 198, 66, 1448, 44, 7, 736226, 5504, 492, 25, 3482516, 35

Offset: 1

Antti Karttunen, May 20 2018

nonn

This is a left inverse of A304755, and also the right inverse if A304755 is a permutation of natural numbers, in which case the fallback clause is unnecessary.

Cf. A304755 (inverse).

PARI
```
\\ Use the program given in A304755.
```

For n >= 1, a(A304755(n)) = n.

A207901 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k, also let R_k equal S_k read in reverse order; then the numbers b_k*R_k are the next 2^k terms.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Feb 21 2012

A permutation of the positive integers (but please note the starting offset: 0-indexed).
This sequence is a variant of A052330.
Shares with A064736, A302350, etc. the property that a(n) is either a divisor or a multiple of a(n+1). - Peter Munn, Apr 11 2018 on SeqFan-list. Note: A302781 is another such "divisor-or-multiple permutation" satisfying the same property. - Antti Karttunen, Apr 14 2018
The offset is 0 since S_0 = {1} denotes the first 2^0 = 1 terms. - Daniel Forgues, Apr 13 2018
This is "Fermi-Dirac piano played with Gray code", as indicated by Peter Munn's Apr 11 2018 formula. Compare also to A303771 and A302783. - Antti Karttunen, May 16 2018

			Start with [1]; appending 2*[1] results in [1,2];
appending 3*[2,1] results in [1,2, 6,3];
appending 4*[3,6,2,1] results in [1,2,6,3, 12,24,8,4];
appending 5*[4,8,24,12,3,6,2,1]
results in [1,2,6,3,12,24,8,4, 20,40,120,60,15,30,10,5];
next append 7*[5,10,30,15,60,120,40,20,4,8,24,12,3,6,2,1],
multiplying by 7 since 6 is already found in the previous terms.
Each new factor is in A050376: [2,3,4,5,7,9,11,13,16,17,19,23,25,29,...].
Continue in this way to generate all the terms of this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..8191 (first 1024 terms from Paul D. Hanna)
Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list
Index entries for sequences that are permutations of the natural numbers

Cf. A001511, A003188, A052330, A050376, A059897, A064706, A302029 (inverse), A302843.
Cf. A064736, A281978, A282291, A302350, A302781, A302783, A303751, A303771, A304085, A304531, A304755 for other divisor-or-multiple permutations or conjectured permutations.
Cf. A302033 (a squarefree analog), A304745.

a = {1}; Do[a = Join[a, Reverse[a]*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

{A050376(n)= local(m, c, k, p); n--; if(n<=0, 2*(n==0), c=0; m=2; while( cA050376(n-1)*Vec(Polrev(A))));A[n]}
for(n=0,63,print1(a(n),",")) \\ edited for offsets by Michel Marcus, Apr 04 2019

up_to_e = 13;
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); };
A003188(n) = bitxor(n, n>>1);
A207901(n) = A052330(A003188(n)); \\ Antti Karttunen, Apr 13 2018

a(n) = A052330(A003188(n)). - Peter Munn, Apr 11 2018
a(n) = A302781(A302843(n)) = A302783(A064706(n)). - Antti Karttunen, Apr 16 2018
a(n+1) = A059897(a(n), A050376(A001511(n+1))). - Peter Munn, Apr 01 2019

Offset changed from 1 to 0 by Antti Karttunen, Apr 13 2018

A303751 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least divisor of a(n-1) not already present, or (if all divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 36, 9, 18, 90, 5, 10, 30, 15, 60, 20, 180, 45, 360, 8, 24, 120, 40, 1080, 27, 54, 270, 135, 540, 108, 2700, 25, 50, 150, 75, 300, 100, 900, 225, 450, 3150, 7, 14, 42, 21, 84, 28, 252, 63, 126, 630, 35, 70, 210, 105, 420, 140, 1260, 315, 2520, 56, 168, 840, 280, 7560, 72, 1800, 200, 600, 4200

Offset: 1

Antti Karttunen, May 01 2018

nonn

The greedy algorithm which constructs this sequence can be understood also in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition {s1+...+sk} via mapping a(n) = prime(s1) * ... * prime(sk), where s1 .. sk are the summands of an integer partition. The choices for constructing the next partition are: If by removing any parts from the partition we can find any smaller partitions that have not already occurred in the sequence, then we choose the one which has the smallest Heinz encoding value. On the other hand, if all partitions obtained by such removals have already occurred in the sequence, then we add to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (A257993), until a partition is found which is not yet in the sequence.
From Antti Karttunen & David A. Corneth, May 01 - 04 2018: (Start)
No two successive descending terms, that is, a(n) > a(n+1) > a(n+2) never occurs.
For n > 1, if a(n) is odd then a(n-1) = 2^h * k * a(n) and a(n+1) = 2^j * a(n) for some h, k and j, that is, odd terms occur between two larger even numbers.
If a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2).
However, when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) might not be a divisor of a(n+2). The first such case occurs at n=64..66, as a(64) = 280 = 2^3 * 5 * 7, a(65) = 7560 = 2^3 * 3^3 * 5 * 7, and a(66) = 72 = 2^3 * 3^2. We have 7560/280 = 27, which is not a divisor of 72 (72/27 = 8/3).
In most cases, when a(n+1) < a(n) then gcd(a(n+1), a(n)/a(n+1)) = 1 (about 87% for the first 100000 descents). However, there are many exceptions to this, the first case occurring at a(65) = 7560 = 2^3 * 3^3 * 5 * 7 and a(66) = 72 =  2^3 * 3^2, with gcd(72,7560/72) = 3.
(End)
From David A. Corneth, May 04 2018: (Start)
The sequence can be partitioned into a tabf sequence with rows having the first element odd and the others even. It would give (1, 2, 6), (3, 12, 4, 36), (9, 18, 90), (5, 10, 30), (15, 60, 20, 180), (45, 360, 8, 24, 120, 40, 1080), (27, 54, 270), ...
It turns out that some rows are multiples of others; for example, the row (5, 10, 30) is five times the row (1, 2, 6). (End)
See also "observed scaling patterns" in the Formula section.
A303750 gives the positions of odd terms.
A282291 and A304531 are unitary divisor variants that satisfy the condition gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1).
The primes 2, 3, 5, 7, 11, 13, 19, 23 and 29 occur at positions 2, 4, 11, 42, 176, 1343, 8470, 57949, 302739, 1632898, thus after 7 and except for 13, a little earlier than they occur in variant A304531.

			a(64) = 280 = 2^3 * 5 * 7 = prime(1)^3 * prime(3) * prime(4), which by Heinz-encoding corresponds to integer partition {1+1+1+3+4}. We try to remove all 1's (to get {3+4}, i.e., prime(3)*prime(4) = 35, but that has already been used as a(52)), or to remove either 3 or 4 or both, but also 8, 40 and 56 have already been used, and if we remove all 1's and either 3 or 4, then also prime(3) and prime(4), 5 and 7 have already been used. So we must add one or more copies of 2 (the least missing part) to find a partition that has not already been used. And it turns out we need to add three copies, to get {1+1+1+2+2+2+3+4} to obtain value prime(1)^3 * prime(2)^3 * prime(3) * prime(4) = 7560 not used before, so a(65) = 7560.
For the next partition, we remove two 2's and both 3 and 4, to get {1+1+1+2+2} which gives Heinz-code 2^3 * 3^2 = 72, which is the smallest divisor of 7560 that has not been used before in the sequence, thus a(66) = 72.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A303752 (inverse).
Cf. A053669, A303750.
Cf. A113552, A282291, A304531, A304755 for similarly defined sequences, and also A064736, A207901, A281978, A302350, A302781, A302783, A303771 for other permutations satisfying the divisor-or-multiple property.
Cf. also A303761.
Cf. A304728, A304729 (see their scatter plots for alternative views to this process).
Differs from a variant A304531 for the first time at n = 66, where a(66) = 72, while A304531(66) = 189.

up_to = 2^14;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303751 = vector(up_to);
m303752 = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m303752,d),v303751[n] = d;mapput(m303752,d,n);break)); if(!v303751[n], p = A053669(prev); while(mapisdefined(m303752,prev), prev *= p); v303751[n] = prev; mapput(m303752,prev,n)); prev = v303751[n]);
A303751(n) = v303751[n];
A303752(n) = mapget(m303752,n);

Observed scaling patterns:
For n =     2 ..     2 +     0, a(n) =  2*a(n-1).
For n =     4 ..     4 +     0, a(n) =  3*a(n-3).
For n =    11 ..    11 +     7, a(n) =  5*a(n-10).
For n =    42 ..    42 +    23, a(n) =  7*a(n-41).
For n =   176 ..   176 +    80, a(n) = 11*a(n-175).
For n =  1343 ..  1343 +   683, a(n) = 13*a(n-1342).
For n =  8470 ..  8470 +  3610, a(n) = 17*a(n-8469).
For n = 57949 .. 57949 + 18554, a(n) = 19*a(n-57948).

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A304757 Restricted growth sequence transform of A046523(A304755(n)).

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A304756 Inverse of A304755: if A304755(k) = n, a(n) = k, or 0 if n does not occur in A304755.

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A207901 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k, also let R_k equal S_k read in reverse order; then the numbers b_k*R_k are the next 2^k terms.

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A303751 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least divisor of a(n-1) not already present, or (if all divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

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