A282291 -id:A282291 - cp's OEIS frontend

A304098 Restricted growth sequence transform of A046523(A282291(n)).

1, 2, 3, 2, 4, 5, 4, 2, 3, 6, 3, 7, 8, 2, 3, 6, 3, 7, 4, 7, 3, 6, 9, 6, 10, 11, 12, 13, 12, 13, 12, 13, 10, 2, 3, 6, 3, 7, 4, 7, 3, 6, 9, 6, 8, 14, 3, 6, 9, 6, 8, 7, 8, 6, 9, 15, 9, 16, 12, 13, 10, 13, 10, 13, 17, 5, 4, 7, 4, 18, 19, 18, 4, 7, 8, 7, 20, 21, 4, 7, 8, 7, 20, 18, 20, 7, 8, 14, 8, 22, 23, 24, 17, 24, 17, 2, 3, 6, 3, 7, 4, 7, 3, 6, 9

Offset: 1

Antti Karttunen, May 17 2018

For all i, j: a(i) = a(j) => A304099(i) = A304099(j).

Antti Karttunen, Table of n, a(n) for n = 1..60824

Cf. A046523, A282291, A304097, A304099.

\\ Needs also code from A282291:
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
write_to_bfile(1,rgs_transform(vector(60824,n,A046523(A282291(n)))),"b304098.txt");

A302853 Suspected permutation of nonnegative integers: a(n) = A052331(A282291(1+n)).

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 12, 8, 9, 11, 10, 14, 30, 16, 17, 19, 18, 22, 20, 28, 24, 25, 27, 26, 31, 5, 7, 15, 13, 29, 21, 23, 87, 64, 65, 67, 66, 70, 68, 76, 72, 73, 75, 74, 78, 94, 80, 81, 83, 82, 86, 84, 92, 88, 89, 91, 90, 95, 69, 71, 79, 77, 93, 85, 117, 32, 33, 41, 40, 44, 36, 52, 48, 49, 57, 56, 60, 124, 96, 97, 105, 104, 108, 100, 116, 112, 113, 121, 120, 125

Offset: 0

Antti Karttunen, May 17 2018

nonn

Shares with sequences like A003188, A006068, A300838, A302846, A303765, A303767, A304083 and A304533 the property that when moving from any a(n) to a(n+1) either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes may occur at the same step.

Cf. A302854 (inverse).
Cf. A052331, A282291.
Cf. also A304533.

up_to_e = 2^15;
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));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A302853(n) = A052331(A282291(1+n)); \\ Needs also code from A282291.

a(n) = A052331(A282291(1+n)).

A304099 Number of prime divisors (counted with multiplicity) in terms of A282291.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 4, 5, 1, 2, 3, 2, 4, 3, 4, 2, 3, 4, 3, 6, 3, 4, 5, 4, 5, 4, 5, 6, 1, 2, 3, 2, 4, 3, 4, 2, 3, 4, 3, 5, 6, 2, 3, 4, 3, 5, 4, 5, 3, 4, 5, 4, 7, 4, 5, 6, 5, 6, 5, 7, 2, 3, 4, 3, 5, 4, 5, 3, 4, 5, 4, 6, 7, 3, 4, 5, 4, 6, 5, 6, 4, 5, 6, 5, 8, 5, 6, 7, 6, 7, 1, 2, 3, 2, 4, 3, 4, 2, 3, 4

Offset: 1

Antti Karttunen, May 17 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..60824

Cf. A001222, A282291, A304098.

PARI
```
A304099(n) = bigomega(A282291(n));
```

a(n) = A001222(A282291(n)).

A282304 a(n) is the least k > 0 such that A282291(n+k) != A282291(n) * A282291(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 2

Rémy Sigrist, Feb 11 2017

nonn

The sequence can be interpreted like this: for any n>1, the b(n) terms of A282291 starting at index n equal the first b(n) terms of A282291, up to a scaling factor of A282291(n).
The presence of huge values in this sequence accounts for the fractal nature of A282291.
The first records in this sequence are:
       n    a(n)  A282291(n)
  ------  ------  ----------
       2       1       2
       8       5       5
      14      11       7
      34      31      11
      96      90      13
     193     185      17
     386     383      19
     770     767      23
    1538    1535      29
    3074    3071      31
   14647   11105      37
   30533   29455      41
   60824   30062      43
  122349   91331      47
  245225  121951      53
  688293  367238      59
The occurrence of a prime number greater than 3 in A282291 seems to set a new record in this sequence.
This sequence has a similar fractal nature as A282291; yet here, repeated portions are identical (not scaled).

Rémy Sigrist, Table of n, a(n) for n = 2..50000

Cf. A282291.

a = {1}; Do[k = 1; While[Or[MemberQ[a, k], Nand[Divisible[#2, #1], CoprimeQ[#1, #2/#1]]] & @@ Sort@ # &@ {k, Last@ a}, k++]; AppendTo[a, k], {n, 300}]; Table[k = 1; While[a[[n + k]] == a[[n]] a[[k + 1]], k++]; k, {n, 2, 120}] (* Michael De Vlieger, Feb 12 2017 *)

A304090 Inverse of A282291: if A282291(k) = n, a(n) = k, or 0 if n does not occur in A282291.

Original entry on oeis.org

1, 2, 4, 6, 8, 3, 14, 26, 66, 9, 34, 5, 96, 15, 11, 6531, 193, 67, 386, 7, 17, 35, 770, 27, 6146, 97, 7484, 19, 1538, 10, 3074, 1210614, 37, 194, 21, 71, 14647, 387, 99, 29, 30533, 16, 60824, 39, 69, 771, 122349, 6532, 367278, 6147, 196, 101, 245225, 7485, 41, 31, 389, 1539, 688293, 12, 1535694, 3075, 73

Offset: 1

Antti Karttunen, May 17 2018

nonn

This is a left inverse of A282291, and also the right inverse if A282291 is a permutation of natural numbers, in which case the fallback-clause is unnecessary.

Cf. A282291, A302854.

PARI
```
\\ Use the program given in A282291.
```

For n >= 1, a(A282291(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

A304531 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least unitary divisor of a(n-1) not already present, or (if all unitary 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, 189, 378, 1890, 945, 3780, 756, 18900, 175, 350, 1050, 525, 2100, 700

Offset: 1

Antti Karttunen, May 14 2018

nonn

The greedy algorithm which constructs the sequence is easiest to grasp in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition. The choices for constructing the next partition are: either remove some parts from the partition, but with the condition that if any summand k is removed, then all copies of k present in partition must be removed in toto. One may remove all copies of several distinct summands as well. If by such a removal of parts we can find any smaller partitions that have not yet 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 one adds to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (see A257993), until a partition is found which is not yet in the sequence. This process also implies that one never removes the summand(s) that was/were just added in the previous step.
It has not yet been rigorously proved that all partitions can be reached this way, i.e., that this sequence is a permutation of natural numbers.
Each a(n+1) is always either a divisor or a multiple of a(n).
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) then (a(n+1) / a(n)) is a divisor of a(n+2). This follows because clearly (in case A) when a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2) because on ascending subsections each successive term is obtained by multiplying by some prime (or its power) not already present. But it is also true (in case B) when a(n) < a(n+1) > a(n+2), as:
In contrast to A303751, this permutation is specified with an additional constraint that gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1). From this then follows that also when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) is guaranteed to be a divisor of a(n+2). It also follows from this that also the squarefree version A304537(n) = A019565(A052331(a(1+n))) satisfies the divisor-or-multiple property.
Odd numbers occur at A304530.
Primes occur at : 2, 4, 11, 42, 237, 1798, 7192, 69611, 431203, 2401568, ...
Primorials (A002110) occur at: 1, 2, 3, 13, 54, 290, 2087, 11333, 118777, 934737, ...

			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 all 1's and the sole 3, to get {2+2+2+4}, with Heinz-encoding prime(2)^3 * prime(4) = 27 * 7 = 189 to obtain the smallest not yet present in sequence, thus a(66) = 189. Note that the partition {1+1+1+2+2} would give even a smaller Heinz-code 2^3 * 3^2 = 72, which also has not been used before, but 72 is not a unitary divisor of 7560, which can be seen from the fact that just one 2 (but not all 2's) was removed from the partition {1+1+1+2+2+2+3+4} that corresponds to 7560. At this point A303751 selects 72 as it has no unitary divisor constraint.

Antti Karttunen, Table of n, a(n) for n = 1..8447
Index entries for sequences that are permutations of the natural numbers

Cf. A304532 (inverse).
Cf. A304530 (positions of odd terms).
Cf. A053669, A215366, A257993, A304533, A304535, A304536, A304537.
Cf. A113552, A282291, A303751 for other variants.
Differs from A303751 for the first time at n=66, where a(66) = 189, while A303751(66) = 72.

up_to = 2^12;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v304531 = vector(up_to);
m304532 = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m304532,d) && (1==gcd(d, prev/d)),v304531[n] = d;mapput(m304532,d,n);break)); if(!v304531[n], p = A053669(prev); while(mapisdefined(m304532,prev), prev *= p); v304531[n] = prev; mapput(m304532,prev,n)); prev = v304531[n]);
A304531(n) = v304531[n];
A304532(n) = mapget(m304532,n);

Observed 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+38, a(n) = 7*a(n-41).
For n = 237 .. 237+64, a(n) = 11*a(n-236).
For n = 1798 .. 1798+336, a(n) = 13*a(n-1797).
For n = 7192 .. 7192+1255, a(n) = 17*a(n-7191).
For n = 69611 .. 69611+4820, a(n) = 19*a(n-69610).
For n = 431203 .. 431203+41802, a(n) = 23*a(n-431202).
For n = 2401568 .. 2401568+131366, a(n) = 29*a(n-2401567).
Derived sequences. For all n >= 1:
A052331(a(n)) = A304533(n-1).
A064547(a(n)) = A304536(n-1).

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).

A113552 Beginning with 1, least divisor of the previous term not included earlier, otherwise the least multiple of the previous term having at least one prime divisor coprime to it and not included earlier.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 20, 5, 10, 30, 15, 60, 420, 7, 14, 42, 21, 84, 28, 140, 35, 70, 210, 105, 630, 9, 18, 90, 45, 180, 36, 252, 63, 126, 1260, 315, 1890, 27, 54, 270, 135, 540, 108, 756, 189, 378, 3780, 945, 5670, 81, 162, 810, 405, 1620, 324, 2268, 567, 1134, 11340, 2835

Offset: 1

Amarnath Murthy, Nov 03 2005

From Michael De Vlieger, May 18 2018: (Start)
In the table below, we note a cycle that exists for n >= 24 and at least through n = 2^14. Let e = floor(n/12). We write multiple k if the condition is false, or the parity of divisor d if d does not occur in a. We can express a(n) as the product of the smallest four primes as shown below.
n (mod 12)   k or d     2     3     5     7
-------------------------------------------
         ODD         3^(e-1)  5     7
         6        2    3^e    5     7
         ODD           3^e
         2        2    3^e
         5        2    3^e    5
         ODD           3^e    5
         4        2^2  3^e    5
         EVEN     2^2  3^e
         7        2^2  3^e          7
         ODD           3^e          7
         2        2    3^e          7
         10       2^2  3^e    5     7
Conjectures:
All terms are divisible only by some combination of the smallest 4 primes; A113553 is finite at four terms.
For n > 24 such that n (mod 12) = 2, a(n) = 3^((n - 2)/12). (End)

			After 4 the next term is 20 and not 8 as 8 and 4 have the same prime divisors.
After a(27) = 18 = 2 * 3^2, the next term a(28) is neither 2*18 = 2^2 * 3^2, nor 3*18 = 2 * 3^3 nor 4*18 = 2^3 * 3^2 as none of them have any prime factors that would not occur already in 18. But 5*18 = 90 has such a factor, and 90 has not occurred before, thus a(28) = 90. - _Antti Karttunen_, May 18 2018

Antti Karttunen & Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Notes on A113552

Differs from A282291 for the first time at n=25, where a(25) = 630, while A282291(25) = 840.

Differs from A304752 for the first time at n=28, where a(28) = 90, while A113552(28) = 72.

S:= {1}: A[1]:= 1:
for n from 2 to 60 do
  d:= min(numtheory:-divisors(A[n-1]) minus S);
  if d < infinity then A[n]:= d
  else
    Q:= numtheory:-factorset(A[n-1]);
    for k from 2 do
      if not member(k*A[n-1], S) and not (numtheory:-factorset(k) subset Q) then
        A[n]:= k*A[n-1];
        break
      fi
    od
  fi;
  S:= S union {A[n]}
od: # Robert Israel, May 22 2018

Nest[Append[#, Block[{d = Complement[Divisors@ #[[-1]], #], k = 2}, If[d != {}, d[[1]], While[Nand[PowerMod[#[[-1]], k, k] != 0, FreeQ[#, k #[[-1]] ] ], k++]; k #[[-1]] ] ] ] &, {1}, 116] (* Robert G. Wilson v, Aug 20 2006, corrected by Michael De Vlieger, May 18 2018 *)

up_to = (2^14)+1;
A007947(n) = factorback(factorint(n)[, 1]);
v113552 = vector(up_to);
m_occurrences = Map();
k=0; prev=1;
for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_occurrences,d),v113552[n] = d;mapput(m_occurrences,d,n);break)); if(!v113552[n], m = 1; try = prev; while(!(prev%A007947(m)) || mapisdefined(m_occurrences,try), m++; try = prev*m); mapput(m_occurrences,v113552[n] = try,n)); prev = v113552[n]);
A113552(n) = v113552[n]; \\ Antti Karttunen, May 18 2018

More terms from Robert G. Wilson v, Aug 20 2006

Sequence's terms changed back to the original author's intended meaning, differing from those computed by Wilson and Johnston from n = 28 onward, with a(28) = 90 instead of a(28) = 72. The latter version was recreated with the new A-number A304752, to which also the old Maple and Mathematica programs were transferred. - Antti Karttunen, May 18 2018

A303771 Divisor-or-multiple permutation of natural numbers, "Fermi-Dirac piano played with May code": a(n) = A052330(A303767(n)).

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 8, 24, 120, 5, 10, 30, 15, 60, 20, 40, 280, 7, 14, 42, 21, 84, 28, 56, 168, 840, 35, 70, 210, 105, 420, 140, 1260, 9, 18, 54, 27, 108, 36, 72, 216, 1080, 45, 90, 270, 135, 540, 180, 360, 2520, 63, 126, 378, 189, 756, 252, 504, 1512, 7560, 315, 630, 1890, 945, 3780, 41580, 11, 22, 66, 33, 132, 44, 88, 264, 1320, 55, 110, 330, 165, 660, 220

Offset: 0

Antti Karttunen, May 02 2018

nonn

Consider A019565. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually primes that are multiplied together. Of course only "squarefree music" (sequences containing only squarefree numbers, A005117) is possible to generate this way, thus we call A019565 a "squarefree piano".
There is a more sophisticated instrument, called "Fermi-Dirac piano" (A052330), with which it is possible to produce sequences that may contain any numbers.
If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to either of these pianos, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). For example, Gray code A003188 and its inverse A006068 are examples of such tapes, and they produce sequences A302033, A207901 and A284003, A302783.
This divisor-or-multiple permutation is obtained by playing "Fermi-Dirac piano" with the same tape which yields A303760 when "squarefree piano" is played with it. Note that A303760 is not a subsequence of this sequence, as its terms occur in different order than the squarefree terms here.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list and comments in A304537.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Michel Marcus, Peter Munn, et al., Discussion on SeqFan-list about similar permutations, April 2018
Index entries for sequences that are permutations of the natural numbers

Cf. A303772 (inverse).
Cf. A019565, A048675, A052330, A303760.
Cf. also A064736, A113552, A207901, A281978, A282291, A302350, A302781, A302783, A303751, A304085, A304531 for similar permutations.

default(parisizemax,2^31);
up_to_e = 16;
up_to = (1 + 2^up_to_e);
v050376 = vector(2+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 == 2+up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303760 = vector(up_to);
m_inverses = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_inverses,d),v303760[n] = d;mapput(m_inverses,d,n);break)); if(!v303760[n], apu = prev; while(mapisdefined(m_inverses,try = prev*A053669(apu)), apu *= A053669(apu)); v303760[n] = try; mapput(m_inverses,try,n)); prev = v303760[n]);
A303760(n) = v303760[n+1];
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A303771(n) = A052330(A048675(A303760(n)));

a(n) = A052330(A303767(n)) = A052330(A048675(A303760(n))). [See comments].

Name amended by Antti Karttunen, May 16 2018

cp's OEIS Frontend

A304098 Restricted growth sequence transform of A046523(A282291(n)).

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A302853 Suspected permutation of nonnegative integers: a(n) = A052331(A282291(1+n)).

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A304099 Number of prime divisors (counted with multiplicity) in terms of A282291.

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A282304 a(n) is the least k > 0 such that A282291(n+k) != A282291(n) * A282291(k+1).

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A304090 Inverse of A282291: if A282291(k) = n, a(n) = k, or 0 if n does not occur in A282291.

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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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