A207901 -id:A207901 - cp's OEIS frontend

A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

1, 2, 6, 3, 24, 12, 4, 8, 120, 60, 20, 5, 40, 10, 30, 15, 840, 420, 140, 35, 7, 280, 70, 14, 210, 105, 21, 168, 84, 28, 56, 7560, 42, 1890, 945, 315, 63, 9, 3780, 1260, 252, 36, 2520, 630, 126, 18, 1512, 756, 189, 27, 378, 54, 1080, 540, 180, 45, 360, 90, 270, 135, 83160, 504, 72, 216, 108, 41580, 13860, 3465, 693, 99, 11, 27720, 6930, 1386, 198, 22, 20790

Offset: 0

Antti Karttunen, May 06 2018

nonn

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

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A304086 (inverse).
Cf. A304083, A304087.
Cf. also A064736, A113552, A207901, A281978, A282291, A302350, A302781, A302783, A303751, A303771 for similar permutations.

up_to_e = 16; \\ Good for computing up to n = (2^16)-1
v050376 = vector(up_to_e);
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));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A304085(n) = A052330(A304083(n)); \\ Needs also code from A304083

a(n) = A052330(A304083(n)).

A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 15, 14, 13, 10, 9, 8, 11, 4, 5, 6, 7, 24, 27, 26, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 21, 20, 23, 40, 43, 42, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 37, 36, 39, 56, 57, 58, 59, 52, 55, 54, 53, 50, 49, 48, 51, 60, 61, 62, 63, 192, 195, 194, 193, 198, 199, 196, 197, 202, 203, 200, 201, 206, 205

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

When A207901, which is a multiplicative walk permutation, is composed from the right with this permutation, the result is A302781, another multiplicative walk permutation.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302843 (inverse).

Cf. A003188, A006068, A057300, A163356, A302846, A302781.

A003188(n) = bitxor(n, n>>1);
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)))); \\ Antti Karttunen, Apr 14 2018
A302844(n) = A003188(A163356(n));

a(n) = A003188(A163356(n)).

a(n) = A006068(A302846(n)).

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

Original entry on oeis.org

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))).

A322017 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, else if A003188(n+1) < A003188(n) then a(n) = n+1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 7, 0, 7, 8, 9, 12, 0, 12, 15, 0, 15, 16, 17, 0, 19, 20, 23, 24, 0, 24, 25, 28, 0, 28, 31, 0, 31, 32, 33, 0, 35, 36, 39, 0, 39, 40, 41, 44, 0, 44, 47, 48, 0, 48, 49, 0, 51, 52, 55, 56, 0, 56, 57, 60, 0, 60, 63, 0, 63, 64, 65, 0, 67, 68, 71, 0, 71, 72, 73, 76, 0, 76, 79, 0, 79, 80, 81, 0, 83, 84, 87, 88, 0, 88, 89

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

For all n, A207901(a(n)) divides A207901(n), and similarly for A302033.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A322015, A322016, A322018.

Cf. also A207901, A302033.

g[n_] := BitXor[n, Floor[n/2]]; a[n_] := If[n == 0, 0, If[g[n] > g[n-1],  n-1, If[g[n+1] < g[n], n+1, 0]]]; Array[a, 100, 0] (* Amiram Eldar, Dec 05 2018 *)

A003188(n) = bitxor(n, n>>1);
A322017(n) = if(0==n, 0, if(A003188(n)>A003188(n-1), n-1, if(A003188(1+n) < A003188(n), n+1, 0)));

A302030 a(n) = 1+A006068(A052331(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 32, 7, 64, 15, 128, 5, 256, 31, 13, 512, 1024, 63, 2048, 9, 29, 127, 4096, 6, 8192, 255, 61, 25, 16384, 14, 32768, 511, 125, 1023, 17, 57, 65536, 2047, 253, 10, 131072, 30, 262144, 121, 49, 4095, 524288, 509, 1048576, 8191, 1021, 249, 2097152, 62, 113, 26, 2045, 16383, 4194304, 12, 8388608, 32767, 33, 505, 241, 126

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

This is the inverse of A207901 if it is considered with a starting offset 1.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

One more than A302029.

Cf. A006068, A052331, A050376, A207901.

up_to_e = 8192;
v050376 = vector(up_to_e);
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); };
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ After code in A006068
A302030(n) = (1+A006068(A052331(n)));

a(n) = 1+A302029(n) = 1+A006068(A052331(n)).

A322018 a(n) = A006068(A129760(A003188(n))).

Original entry on oeis.org

0, 0, 3, 0, 7, 4, 7, 0, 15, 8, 11, 8, 15, 12, 15, 0, 31, 16, 19, 16, 23, 20, 23, 16, 31, 24, 27, 24, 31, 28, 31, 0, 63, 32, 35, 32, 39, 36, 39, 32, 47, 40, 43, 40, 47, 44, 47, 32, 63, 48, 51, 48, 55, 52, 55, 48, 63, 56, 59, 56, 63, 60, 63, 0, 127, 64, 67, 64, 71, 68, 71, 64, 79, 72, 75, 72, 79, 76, 79, 64, 95, 80, 83, 80, 87, 84, 87, 80

Offset: 0

Antti Karttunen, Nov 27 2018

nonn

For all n, A207901(a(n)) divides A207901(n), and similarly for A302033.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A006068, A035327, A129760.

Cf. also A207901, A302033, A322017.

A003188(n) = bitxor(n, n>>1);
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ After code in A006068
A129760(n) = bitand(n,n-1); \\ From A129760
A322018(n) = A006068(A129760(A003188(n)));

a(n) = A006068(A129760(A003188(n))).

cp's OEIS Frontend

A304085 Divisor-or-multiple permutation of natural numbers: a(n) = A052330(A304083(n)).

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A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

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

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A322017 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, else if A003188(n+1) < A003188(n) then a(n) = n+1, otherwise a(n) = 0.

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A302030 a(n) = 1+A006068(A052331(n)).

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A322018 a(n) = A006068(A129760(A003188(n))).

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