A003714 -id:A003714 - cp's OEIS frontend

A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

1, 2, 2, 6, 2, 30, 2, 60, 10, 42, 2, 4200, 2, 126, 70, 660, 2, 9240, 2, 13860, 210, 330, 2, 5082000, 14, 78, 220, 32760, 2, 3783780, 2, 42900, 550, 780, 294, 924924000, 2, 1092, 130, 41621580, 2, 3898440, 2, 112200, 60060, 306, 2, 28078050000, 42, 235620, 1300, 92820, 2, 200119920, 770, 128648520, 1820, 1122, 2, 424964656116000, 2, 3366

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3194

Cf. A003714, A019565, A300835 (rgs-transform of this sequence), A300836.

Cf. also A293214, A293216, A293221, A293222, A293231.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };

a(n) = Product_{d|n, dA019565(A003714(d)).

For n >= 1, A001222(a(n)) = A300836(n).

A345201 Bit-reverse the odd part of the Zeckendorf representation of n: a(n) = A022290(A057889(A003714(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 24, 30, 26, 27, 31, 29, 25, 28, 32, 33, 34, 35, 36, 37, 48, 39, 43, 49, 42, 40, 44, 50, 53, 47, 38, 41, 45, 51, 52, 46, 54, 55, 56, 57, 58, 77, 60, 69, 78, 63, 64, 70, 79, 85

Offset: 0

Rémy Sigrist, Jun 10 2021

This sequence is a self-inverse permutation of the nonnegative integers.

This sequence is similar to A343150 and to A344682.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program for A345201
Rémy Sigrist, Alternate PARI program for A345201
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A022290, A057889, A343150, A344682.

PARI
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See Links section.
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a(n) < A000045(k) for any n < A000045(k).

A129761 First differences of Fibbinary numbers (A003714).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 22, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 43, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 86, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 22, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 171, 1, 1, 2, 1, 3, 1, 1, 6

Offset: 0

Ralf Stephan, May 14 2007

nonn

Theorem: If the Zeckendorf representation of M ends with exactly k >= 0 zeros, ...10^k, then a(n) = ceiling(2^k/3). Also, if the Zeckendorf representation of n (A014417(n)) is even then a(n) is given by A319952, otherwise a(n) = 1. - Jeffrey Shallit and N. J. A. Sloane, Oct 03 2018

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (First 2500 terms from Vincenzo Librandi)

Cf. A000045, A000071, A003714, A005578, A035614, A319432.

with(combinat): F:=fibonacci:
A072649:= proc(n) local j; global F; for j from ilog[(1+sqrt(5))/2](n)
       while F(j+1)<=n do od; (j-1); end:
A003714 := proc(n) global F; option remember; if(n < 3) then RETURN(n); else RETURN((2^(A072649(n)-1))+A003714(n-F(1+A072649(n)))); fi; end:
A129761 := n -> A003714(n+1)-A003714(n):
[seq(A129761(n),n=0..120)]; # N. J. A. Sloane, Oct 03 2018, borrowing programs from other sequences

Differences[Select[Range[600], !MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &]] (* Harvey P. Dale, Jul 17 2011 *)

a(n) = A005578(A035614(n)). - Alan Michael Gómez Calderón, Nov 01 2023

a(0)=1 added by N. J. A. Sloane, Oct 02 2018

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 7, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 41, 62, 63, 64, 2, 65, 66, 67, 68, 69

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

For all i, j: a(i) = a(j) => A300836(i) = A300836(j).

			For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).

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

Cf. A003714, A014417, A019565, A300834, A300836.

Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.

up_to = 65537;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Antti Karttunen, Aug 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A003714, A019565, A072649, A318464, A318465.

Cf. also A293442, A293443, A300834, A318470.

A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318464(n).

A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 4, 5, 2, 4, 2, 5, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 1, 2, 9, 10, 4, 8, 4, 5, 8, 9, 4, 10, 5, 10, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 1, 2, 17, 18, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 2, 4, 18, 20, 2, 5, 18, 21, 8, 16, 8, 9, 16, 17

Offset: 0

Rémy Sigrist, Jul 06 2024

In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  -----------
   0  0
   1  0, 1
   2  0, 2
   3  1, 2
   4  0, 4
   5  0, 1, 4, 5
   6  2, 4
   7  2, 5
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  1, 2, 9, 10
  12  4, 8
  13  4, 5, 8, 9
  14  4, 10
  15  5, 10
  16  0, 16

Rémy Sigrist, Table of n, a(n) for n = 0..8118 (rows for n = 0..1023 flattened)
Index entries for sequences related to binary expansion of n

See A295989 and A374361 for similar sequences.

Cf. A003714, A037800, A277561, A374355, A374356.

row(n) = { my (r = [0], e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], r = concat([v + y | v <- r], [v + x | v <- r]); break;););); return (r); }

T(n, 0) = 0 iff n is a fibbinary number.
T(n, k) + T(n, A277561(n)-1-k) = n.
T(n, 0) = A374355(n).
T(n, A277561(n)-1) = A374356(n).
Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).

A374355 a(n) is the least fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0, 0, 0, 1, 0, 0, 2, 2, 8, 8, 8, 9, 8, 8, 10, 10, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 16, 16, 16, 17, 16, 16, 18, 18, 16, 16, 16, 17, 20, 20, 20, 21, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): replace every other bit with zero (starting with the first bit) in each run of consecutive 1's in the binary expansion of n.

			The first terms, in binary and in decimal, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     0     101          0
   6     2     110         10
   7     2     111         10
   8     0    1000          0
   9     0    1001          0
  10     0    1010          0
  11     1    1011          1
  12     4    1100        100
  13     4    1101        100
  14     4    1110        100
  15     5    1111        101
  16     0   10000          0

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003714, A374354, A374356.

a(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += y; break;););); return (v); }

a(n) = A374354(n, 0).
a(n) = n - A374356(n).
a(n) >= 0 with equality iff n is a fibbinary number.

A277332 a(n) = A253565(A003714(n)).

Original entry on oeis.org

1, 2, 3, 5, 9, 7, 25, 15, 11, 49, 35, 21, 75, 13, 121, 77, 55, 245, 33, 147, 105, 17, 169, 143, 91, 847, 65, 605, 385, 39, 363, 231, 165, 735, 19, 289, 221, 187, 1859, 119, 1183, 1001, 85, 845, 715, 455, 4235, 51, 507, 429, 273, 2541, 195, 1815, 1155, 23, 361, 323, 247, 3757, 209, 3179, 2431, 133, 2023, 1547, 1309, 13013, 95

Offset: 0

Antti Karttunen, Oct 12 2016

nonn

After the initial terms 1, 2 and 3, all other terms can be inductively generated by applying any finite composition-combination of A253560 and A253550 to 3, but with such a restriction that A253560 may not be applied twice in succession.
A permutation of A277334.
Note how A253565(A022340(n)) = A253565(2*A003714(n)) yields a permutation of A056911, odd squarefree numbers.

			55 = A253550(A253550(A253560(A253550(3)))), 55 is in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A003714, A022340, A253550, A253560, A253565.
Cf. A277334 (same sequence sorted into ascending order).
Cf. also A056911, A277006, A277331.

(define (A277332 n) (A253565 (A003714 n)))

a(n) = A253565(A003714(n)).

A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 5, 3, 11, 1, 1, 1, 7, 1, 1, 3, 3, 3, 13, 5, 3, 3, 5, 11, 11, 1, 1, 1, 39, 1, 1, 7, 7, 1, 1, 1, 3, 3, 13, 3, 7, 3, 21, 13, 23, 5, 5, 3, 3, 3, 9, 5, 11, 11, 9, 11, 43, 1, 1, 1, 35, 1, 1, 39, 15, 1, 1, 1, 31, 7, 57, 7, 7, 1

Offset: 0

Rémy Sigrist, Mar 14 2018

This sequence is well defined: for any positive n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides f = abs(A195156(i) - A195156(j)), and as f is a Fibbinary number, a(n) <= f/n.

All terms are odd.

			The first terms, alongside the binary representation of n * a(n), are:
  n  a(n)   bin(n * a(n))
  -- ----   -------------
   0    1           0
   1    1           1
   2    1          10
   3    3        1001
   4    1         100
   5    1         101
   6    3       10010
   7    3       10101
   8    1        1000
   9    1        1001
  10    1        1010
  11    3      100001
  12    3      100100
  13    5     1000001
  14    3      101010
  15   11    10100101
  16    1       10000
  17    1       10001
  18    1       10010
  19    7    10000101
  20    1       10100

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A070939(n * a(n)))

Cf. A003714, A070939, A195156, A300889.

a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k)))

a(n) = A300889(n) / n for any n > 0.
a(2*n) = a(n).
a(n) = 1 iff n belongs to A003714.

A300889 a(n) is the least positive multiple of n which is a Fibbinary number (A003714).

Original entry on oeis.org

1, 2, 9, 4, 5, 18, 21, 8, 9, 10, 33, 36, 65, 42, 165, 16, 17, 18, 133, 20, 21, 66, 69, 72, 325, 130, 81, 84, 145, 330, 341, 32, 33, 34, 1365, 36, 37, 266, 273, 40, 41, 42, 129, 132, 585, 138, 329, 144, 1029, 650, 1173, 260, 265, 162, 165, 168, 513, 290, 649

Offset: 1

Rémy Sigrist, Mar 14 2018

			The first terms, alongside their binary representation, are:
  n  a(n)    binary(a(n))
  -- ----    ------------
   1    1           1
   2    2          10
   3    9        1001
   4    4         100
   5    5         101
   6   18       10010
   7   21       10101
   8    8        1000
   9    9        1001
  10   10        1010
  11   33      100001
  12   36      100100
  13   65     1000001
  14   42      101010
  15  165    10100101
  16   16       10000
  17   17       10001
  18   18       10010
  19  133    10000101
  20   20       10100

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A003714, A300867.

a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k*n)))

a(n) = n * A300867(n).
a(2*n) = 2*a(n).
a(n) = n iff n belongs to A003714.

cp's OEIS Frontend

A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

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A345201 Bit-reverse the odd part of the Zeckendorf representation of n: a(n) = A022290(A057889(A003714(n))).

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A129761 First differences of Fibbinary numbers (A003714).

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A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

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A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

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A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

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A374355 a(n) is the least fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

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A277332 a(n) = A253565(A003714(n)).

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