A061446 -id:A061446 - cp's OEIS frontend

A105602 Divide each Fibonacci number by its primitive part.

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 24, 1, 13, 10, 21, 1, 136, 1, 165, 26, 89, 1, 1008, 5, 233, 34, 1131, 1, 26840, 1, 987, 178, 1597, 65, 46512, 1, 4181, 466, 47355, 1, 1269736, 1, 53133, 10370, 28657, 1, 2179296, 13, 825275, 3194, 364179, 1, 14927768, 445

Offset: 1

Paul Barry, Apr 15 2005

Sylvester dividends for Fibonacci numbers.

a(n)=1 for n=1, 4 and all primes, which is sequence A046022.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000045, A046022, A061446, A126015.

a[n_] := Fibonacci[n]/Product[Fibonacci[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n],{n,55}] (* James C. McMahon, Jan 25 2024 *)

a(n) = Fibonacci(n)/A061446(n).

A105603 Sylvester-Jacobsthal cyclotomic numbers.

Original entry on oeis.org

1, 1, 3, 5, 11, 7, 43, 17, 57, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 205, 5419, 2047, 2796203, 241, 1016801, 8191, 261633, 3277, 178956971, 151, 715827883, 65537, 1397419, 131071, 24214051, 4033, 45812984491, 524287, 22366891, 61681

Offset: 1

Paul Barry, Apr 15 2005

Primitive parts of Jacobsthal numbers A001045.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.

Cf. A020501, A001045, A061446 (with references), A008555.

a(n)=product{k=1..n-1, if(gcd(n, k)=1, 2+exp(2*pi*I*k/n), 1)}, I=sqrt(-1)

a(n) = (-1)^phi(n)*cyclotomic(n, -2) for n >= 2 (for n = 1 this would be 3), with phi(n) = A000010(n). - Wolfdieter Lang, Jan 19 2015

A178764 Ratio of the primitive part of Fibonacci(n) to the product of primitive prime factors of Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

T. D. Noe, Jun 10 2010

nonn

Except for a(6), it can be shown that all terms are either 1 or prime. In the first 10^5 terms, only 151 are greater than 1.

T. D. Noe, Table of n, a(n) for n=1..10000
Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci Integers (preprint)

a(n) = A061446(n) / A178763(n).

A250269 Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

Original entry on oeis.org

1, 2, 6, 12, 120, 60, 5040, 1680, 60480, 15120, 39916800, 55440, 6227020800, 8648640, 1816214400, 518918400, 355687428096000, 147026880, 121645100408832000, 55870214400, 1689515283456000, 14079294028800, 25852016738884976640000, 771008958720

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

The title is analogous to the title of A061446.
For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence.  Not every divisibility sequence b corresponds to some integer sequence a such that b(n) = Product_{d|n} a(d), however.
This sequence is not itself a divisibility sequence; a(15) does not divide a(30), for example.

			The divisors of 10 are 1, 2, 5 and 10. 10! = a(1) * a(2) * a(5) * a(10) = 1 * 2 * 120 * 15120 = 3628800.
Between 1 and 10 inclusive, 4 integers are coprime to 10: 1, 3, 7 and 9. Let b(n) = lcm (1...n) = A003418(n), and let [x] denote the floor function. Then:
a(10) = b[10/1] * b[10/3] * b[10/7] * b[10/9]
"   "   = b(10) * b(3) * b(1) * b(1)
"   "   = 2520 * 6 * 1 * 1
"   "   = 15120.

Michael De Vlieger, Table of n, a(n) for n = 1..456
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.

Cf. A000142, A075071. Subsequence of A250270.

Cf. A000010 (comments on product formulas), A008683.

Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* Michael De Vlieger, Nov 11 2021 *)

a(n)={my(r=1);fordiv(n,d,r*=d!^moebius(n/d));r} \\ Joerg Arndt, Jan 18 2015

a(n) = Product_{i = 1..n, gcd(n, i) = 1} lcm (1..floor(n/i)).
a(n) = Product_{i = 1..floor(n/2), gcd(n, i) = 1} lcm (1..floor(n/i)) (equivalent formula).
a(n) = n! iff n is prime.
a(n) = Product_{d|n} (d!)^moebius(n/d). - Joerg Arndt, Jan 18 2015
a(n) = Product_{k=1..n} (gcd(n,k)!)^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} ((n/gcd(n,k))!)^(mu(gcd(n,k))/phi(n/gcd(n,k))) where mu = A008683, phi = A000010. - Richard L. Ollerton, Nov 08 2021

A265575 LCM-transform of Euler totient numbers (A000010).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 11, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 2, 1, 13, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 41, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..26003
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.

Cf. A000010.

Other LCM-transforms are A061446, A265574, A265576, A265577, A265578.

LCMXfm:=proc(a) local L,i,n,g,b;
L:=nops(a);
g:=Array(1..L,0); b:=Array(1..L,0);
b[1]:=a[1]; g[1]:=a[1];
for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
lprint([seq(b[i],i=1..L)]);
end;
with(numtheory);
t1:=[seq(phi(n),n=1..100)];
LCMXfm(t1);

LCMXfm[a_List] := Module[{L = Length[a], b, g}, b[1] = g[1] = a[[1]]; b[] = 0; g[] = 0; Do[g[n] = LCM[g[n - 1], a[[n]]]; b[n] = g[n]/g[n - 1], {n, 2, L}]; Array[b, L]];
LCMXfm[Table[EulerPhi[n], {n, 1, 100}]] (* Jean-François Alcover, Dec 05 2017, from Maple *)

up_to = 10000;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
v265575 = LCMtransform(vector(up_to,i,eulerphi(i)));
A265575(n) = v265575[n]; \\ Antti Karttunen, Nov 09 2018

A061488 Factorize the Fibonacci numbers in order, skipping F(0)-F(2), F(6)=8 and F(12)=144; at each step at least one new prime will occur; append to the sequence the smallest such new prime.

Original entry on oeis.org

2, 3, 5, 13, 7, 17, 11, 89, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101, 6376021

Offset: 3

N. J. A. Sloane, Nov 08 2001

Carmichael showed that the sequence is well defined.
Same as A001578 without the "1" terms.
Given the definition, in particular omission of F(6) and F(12), setting offset=1 would be more adequate; offset=5 (= number of omitted terms) would give A001578 for n > 12 on. - M. F. Hasler, Oct 21 2012

T. D. Noe, Table of n, a(n) for n=3..998
Ron Knott, Mathematics of the Fibonacci Series
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences

Cf. A000045, A061446.

f[n_] := Block[{p = First /@ FactorInteger[ Fibonacci[ n]]}, k = 1; lmt = 1 + Length@ p; While[k < lmt && MemberQ[lst, p[[k]]], k++]; If[k < lmt, AppendTo[lst, p[[k]]]]]; lst = {}; Do[ f[n], {n, 3, 51}]; lst (* Robert G. Wilson v, Oct 23 2012 *)

a(n) = A001578(n+2) from n=11 on. - M. F. Hasler, Oct 21 2012

More terms from Vladeta Jovovic and Lior Manor, Nov 09 2001

Corrected by T. D. Noe, Feb 10 2007

A126025 Number of mappings f:{1,2,3,...,n} -> {1,2,3,...,n} such that gcd(f(x),f(y)) = f(gcd(x,y)) for all x,y in {1,2,3,...,n}.

Original entry on oeis.org

1, 3, 9, 26, 106, 191, 954, 2427, 8404, 15945, 111952, 141117, 1176623, 2270566, 4477947, 10345290, 104257447, 145407966, 1633452518, 2517488363, 5024167821, 9148333241, 120260250853

Offset: 1

John W. Layman, Feb 27 2007

The greatest common divisor condition was suggested by A061446.

Manfred Scheucher, Sage Script
Manfred Scheucher, C Code

Cf. A061446.

Cf. A000312.

a126025 n = h n1s 0 where
   h us c = if us == nns then c + 1 else h (succ us) (c + g) where
     g = if and [f x `gcd` f y == f (x `gcd` y) |
                 x <- [1 .. n - 1], y <- [x + 1 .. n]] then 1 else 0
     f = (us !!) . subtract 1
   succ (z:zs) = if z < n then (z + 1) : zs else 1 : succ zs
   n1s = take n [1, 1 ..]; nns = take n [n, n ..]
-- Reinhard Zumkeller, May 04 2014

a(10)-a(22) from Manfred Scheucher, Jun 06 2015

a(23) from Manfred Scheucher, Aug 13 2015

A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).

Original entry on oeis.org

27, 807, 1707, 2977, 3027, 3277, 4717, 5137, 5677, 5917, 5967, 6187, 7087, 7357, 7597, 7707, 8217, 9117, 9297, 9387, 9667, 9877, 9927, 9997, 10387, 11097, 11647, 11797, 12727, 13407, 13867, 15757, 15987, 16327, 16507, 16857, 17347, 17767, 18237, 18817, 18997

Offset: 1

Arkadiusz Wesolowski, Apr 06 2009

nonn

Chris Caldwell, The Prime Glossary, Fibonacci number

Subsequence of A159259.

Cf. A000045, A023172, A061446, A159231, A181890.

lst = {1}; Do[f = Fibonacci[a]; Do[f = f/GCD[f, lst[[d]]], {d, Most[Divisors[a]]}]; AppendTo[lst, f], {a, 2, 19000}]; Flatten[Table[If[! PrimeQ[n] && Mod[lst[[n]], 8*n^2 - 2*n - 1] == 0, n, {}], {n, 19000}]] (* Arkadiusz Wesolowski, Dec 12 2011 *)

A253807 Primitive part of A006190(n), n >= 1.

Original entry on oeis.org

1, 3, 10, 11, 109, 12, 1189, 119, 1297, 131, 141481, 118, 1543321, 1429, 15445, 14159, 183642229, 1299, 2003229469, 14041, 1837837, 170039, 238367471761, 14158, 23854956949, 1854841, 2186871697, 1670761, 309400794703549

Offset: 1

Wolfdieter Lang, Jan 19 2015

A006190(n) = Product_{k divides n} a(k), n >= 1.

Cf. A000010, A006190, A008683, A013595.

Cf. A061446, A008555, A105603.

(* b = A006190 *) b[0] = 0; b[1] = 1; b[n_] := b[n] = 3*b[n-1] + b[n-2]; a[n_] := Product[b[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 20 2015 *)

a(n) = ((3-sqrt(13))/2)^phi(n)*cyclotomic(n, -(11 - 3*sqrt13)/2) for n >= 1 and a(1) = 1, where phi is Euler's totient A000010 and the coefficient table for the cyclotomic polynomials is given in A013595.

a(n) = Product_{d|n} A006190(d)^mu(n/d), where mu = A008683, n >= 1.

A265577 LCM-transform of Yellowstone permutation A098550.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 5, 7, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 3, 11, 13, 1, 1, 1, 1, 1, 1, 17, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 19, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 3, 1, 29, 31, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jan 02 2016

nonn

A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.

Cf. A064413.

Other LCM-transforms are A061446, A265574, A265575, A265576.

LCMXfm:=proc(a) local L,i,n,g,b;
L:=nops(a);
g:=Array(1..L,0); b:=Array(1..L,0);
b[1]:=a[1]; g[1]:=a[1];
for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
lprint([seq(b[i],i=1..L)]);
end;
# let t1 contain the first 100 terms of A098550
LCMXfm(t1);

LCMXfm[a_List] := Module[{L = Length[a], b, g}, b[1] = g[1] = a[[1]]; b[] = 0; g[] = 0; Do[g[n] = LCM[g[n-1], a[[n]]]; b[n] = g[n]/g[n-1], {n, 2, L}]; Array[b, L]];
y[n_ /; n <= 3] := n; y[n_] := y[n] = For[k = 1, True, k++, If[ FreeQ[ Array[y, n-1], k], If[GCD[k, y[n-1]] == 1 && GCD[k, y[n-2]] > 1, Return[k]]]];
Yperm = Array[y, 100];
LCMXfm[Yperm] (* Jean-François Alcover, Dec 03 2017 *)

cp's OEIS Frontend

A105602 Divide each Fibonacci number by its primitive part.

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A105603 Sylvester-Jacobsthal cyclotomic numbers.

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A178764 Ratio of the primitive part of Fibonacci(n) to the product of primitive prime factors of Fibonacci(n).

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A250269 Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

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A265575 LCM-transform of Euler totient numbers (A000010).

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Maple

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A061488 Factorize the Fibonacci numbers in order, skipping F(0)-F(2), F(6)=8 and F(12)=144; at each step at least one new prime will occur; append to the sequence the smallest such new prime.

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A126025 Number of mappings f:{1,2,3,...,n} -> {1,2,3,...,n} such that gcd(f(x),f(y)) = f(gcd(x,y)) for all x,y in {1,2,3,...,n}.

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Haskell

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A159234 Composite numbers n such that 8*n^2-2*n-1 divides the primitive part U(n) of Fibonacci(n).

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Mathematica

A253807 Primitive part of A006190(n), n >= 1.

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A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).