A304085 -id:A304085 - cp's OEIS frontend

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.

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

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

A304083 Permutation of nonnegative integers: Minimal subset/superset bitmask transform of A054429.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 8, 13, 9, 11, 10, 31, 30, 28, 24, 16, 29, 25, 17, 27, 26, 18, 23, 22, 20, 21, 63, 19, 59, 58, 56, 48, 32, 62, 60, 52, 36, 61, 57, 49, 33, 55, 54, 50, 34, 51, 35, 47, 46, 44, 40, 45, 41, 43, 42, 127, 53, 37, 39, 38, 126, 124, 120, 112, 96, 64, 125, 121, 113, 97, 65, 123, 122, 114, 98, 66, 119, 118, 116, 100, 68, 117, 101

Offset: 0

Antti Karttunen, May 06 2018

In "minimal subset/superset bitmask transform", applicable to any N -> N injection f, we start from a(0) = 0, after which for n > 0, if there are one or more k_i that are not already present in the sequence among terms a(0) .. a(n-1), and for which bitor(k_i,a(n-1)) = a(n-1), then a(n) = that k_i for which f(k_i) is minimized; otherwise, a(n) = that h_i for which f(h_i) is minimized among the infinite set of numbers h_i for which bitand(h_i,a(n-1)) = a(n-1) and that are not yet present in the sequence. In this case f(n) = A054429(n).

Shares with permutations like A003188, A006068, A163252, A300838, A302846, A303763, A303765, A303767, A303773 and A303775 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. Note that A303767 is obtained when the same transform is applied to A001477, and A303775 when it is applied to A193231.

			After a(3) = 2, "10" in binary, there are no submasks that wouldn't have been used, so one selects from supermasks h_i = "110" (6), "111" (7), "1010" (10), "1011" (11), "1110" (14), "1111" (15), "10010" (18), "10011" (19), etc. that one for which A054429(h_i) is minimized, which happens to be at 6 (as A054429(6) = 5, but A054429(7) = 4, and for n >= 8, A054429(n) >= 8), thus a(4) = 7.
After a(4) = 7, "111" in binary, the submasks "1", "10", and "11" (1-3) are already present in sequence, while submasks "100", "101", "110" (4-6) are not present, and because A054429 is minimized on these three at 6, a(5) = 6.

Cf. A304084 (inverse).
Cf. A054429.
Cf. also A303767, A303775, A304085, A304087, A304088, A304089.

allocatemem(2^30);
default(parisizemax,2^31);
up_to = (2^17)+2;
A054429(n) = ((3<<#binary(n\2))-n-1);
find_minimal_submask_for_A054429(n,m_inverses) = { my(minval=0,minmask=0); for(m=1,n,if((bitor(m,n)==n) && !mapisdefined(m_inverses,m) && (!minval || (A054429(m) < minval)), minval = A054429(m); minmask = m)); (minmask); };
find_minimal_supermask_for_A054429(n,m_inverses) = { my(minval=0,minmask=0); for(m=1,(1<<(1+#binary(n)))-1,if((bitand(m,n)==n) && !mapisdefined(m_inverses,m) && (!minval || (A054429(m) < minval)), minval = A054429(m); minmask = m)); (minmask); };
v304083 = vector(up_to);
m304084 = Map();
w=1; for(n=1,up_to,s = Set([]); if((submask = find_minimal_submask_for_A054429(w,m304084)), w = submask, w = find_minimal_supermask_for_A054429(w,m304084)); v304083[n] = w; mapput(m304084,w,n));
A304083(n) = if(!n,n,v304083[n]);
A304084(n) = if(!n,n,mapget(m304084,n));

Derived sequences:
A052330(a(n)) = A304085(n).
A019565(a(n)) = A304087(n).
A000120(a(n)) = A304089(n).

A304087 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304083(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 7, 70, 14, 42, 21, 2310, 1155, 385, 77, 11, 770, 154, 22, 462, 231, 33, 330, 165, 55, 110, 30030, 66, 6006, 3003, 1001, 143, 13, 15015, 5005, 715, 65, 10010, 2002, 286, 26, 4290, 2145, 429, 39, 858, 78, 2730, 1365, 455, 91, 910, 182, 546, 273, 510510, 1430, 130, 390, 195, 255255, 85085, 17017, 2431, 221, 17, 170170

Offset: 0

Antti Karttunen, May 06 2018

nonn

Each a(n+1) is either a divisor or a multiple of a(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A019565, A304083, A304085.

Cf. also A303778.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304087(n) = A019565(A304083(n)); \\ Needs also code from A304083.

a(n) = A019565(A304083(n)).

cp's OEIS Frontend

A304086 Inverse of A304085.

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

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A304083 Permutation of nonnegative integers: Minimal subset/superset bitmask transform of A054429.

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A304087 Divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304083(n)).

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