A304533 -id:A304533 - cp's OEIS frontend

A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

1, 2, 6, 3, 15, 5, 65, 13, 26, 182, 7, 14, 42, 21, 105, 35, 455, 91, 910, 10, 30, 210, 70, 2730, 39, 78, 546, 273, 1365, 195, 7995, 41, 82, 246, 123, 615, 205, 2665, 533, 1066, 11726, 11, 22, 66, 33, 165, 55, 715, 143, 286, 2002, 77, 154, 462, 231, 1155, 385, 5005, 1001, 10010, 110, 330, 2310, 770, 30030, 429, 858, 6006, 3003, 15015, 2145, 87945, 451, 902

Offset: 0

Antti Karttunen, May 15 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).
Consider A052330. 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 "Fermi-Dirac primes" (A050376) that are multiplied together.
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 this piano, 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). Furthermore, if the given sequence p is itself a permutation of natural numbers, then also the produced sequence is. For example, Gray code A003188 and its inverse A006068 are such sequences, and when given as an "input tape" for A052330, they produce permutations A207901 and A302783.
There is a simpler instrument, called "squarefree piano" (A019565), with which it is possible to produce similar divisor-or-multiple sequences, but that contain only squarefree numbers. Given A003188 or A006068 as an input tape for it produces correspondingly sequences A302033 and A284003.
This sequence is obtained by playing "squarefree piano" with the same tape which yields A304531 when "Fermi-Dirac piano" is played with it. However, in this case the sequence A304531 is produced by a greedy algorithm, and thus its tape (A304533) is actually a back-formation, obtained from the "music" (A304531) by applying "tape-recorder" (A052331) to it. Note that this in not a subsequence of A304531, as the terms occur in different order than the squarefree terms of A304531.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list.

Antti Karttunen, Table of n, a(n) for n = 0..8447
Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list, April 2018

Cf. A019565, A304531, A304533.

Cf. also A303760, A303771.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304533(n) = A052331(A304531(1+n));
A304537(n) = A019565(A304533(n));

a(n) = A019565(A304533(n)) = A019565(A052331(A304531(1+n))).

A304534 Inverse of A304533.

Original entry on oeis.org

0, 1, 3, 2, 5, 19, 4, 20, 10, 11, 13, 12, 15, 22, 14, 21, 41, 42, 44, 43, 46, 60, 45, 61, 51, 52, 54, 53, 56, 63, 55, 62, 7, 8, 24, 25, 6, 81, 29, 88, 17, 9, 27, 26, 16, 18, 28, 23, 48, 49, 65, 66, 47, 157, 70, 301, 58, 50, 68, 67, 57, 59, 69, 64, 236, 237, 239, 238, 241, 255, 240, 256, 246, 247, 249, 248, 251, 258, 250, 257, 277, 278, 280, 279, 282, 296, 281

Offset: 0

Antti Karttunen, May 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10116
Index entries for sequences that are permutations of the natural numbers

Cf. A304533 (inverse).

Cf. A052330, A304531, A304532.

For all n >= 0, a(A304533(n)) = n.

a(n) = A304532(A052330(n))-1. [This formula works if A304531 and thus A304533 are indeed permutations, containing all natural numbers.]

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).

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)).

cp's OEIS Frontend

A304535 Restricted growth sequence transform of A278222(A304533(n)).

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A304536 Binary weight of terms of A304533; Number of terms of A050376 in "Fermi-Dirac factorization" of A304531(1+n).

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A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

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A304534 Inverse of A304533.

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

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