A061446 -id:A061446 - cp's OEIS frontend

A265578 LCM-transform of number of divisors function (A000005).

1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 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, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jan 02 2016

nonn

Terms larger than one occur at n = 2, 4, 6, 16, 24, 36, 64, 120, 840, 900, 1296, 7560, 44100, 46656, 83160, ... - Antti Karttunen, Nov 06 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
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.
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000005.

Other LCM-transforms are A061446, A265574, A265575, A265576, A265577.

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(tau(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[DivisorSigma[0, n], {n, 1, 100}]] (* Jean-François Alcover, Dec 05 2017, from Maple *)

up_to = 16384;
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); };
v265578 = LCMtransform(vector(up_to,i,numdiv(i)));
A265578(n) = v265578[n]; \\ Antti Karttunen, Nov 06 2018

A306825 Primitive part of A001353(n).

Original entry on oeis.org

1, 4, 15, 14, 209, 13, 2911, 194, 2703, 181, 564719, 193, 7865521, 2521, 34945, 37634, 1525870529, 2701, 21252634831, 37441, 6779137, 489061, 4122901604639, 37633, 274758906449, 6811741, 19726764303, 7263361, 11140078609864049, 40321, 155161278879431551

Offset: 1

Jianing Song, Mar 16 2019

nonn

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of 1/q is equal to that of 1/p. If q = a(2p) = A001353(2*p)/(4*A001353(p)) = ((2 + sqrt(3))^p + (2 - sqrt(3))^p)/4 is prime (this happens for p = 3, 5, 7, 11, 13, 17, 19, 79, 151, 199, 233, 251, 317, ...), where p is an odd prime, then q is a unique-period prime in base b = (sqrt(12*q^2 - 3) - 1)/2 (1/q has period length 3) as well as in base b' = (sqrt(12*q^2 - 3) + 1)/2 (1/q has period length 6). For example, a(6) = 13 is prime, so 13 is the only prime whose reciprocal has period length 3 in base 22 and the only prime whose reciprocal has period length 6 in base 23. Compare: If q = A000129(p) = A008555(p), then q is a unique-period prime in base b = sqrt(2*q^2 - 1) (1/q has period length 4).
By Lucas-Lehmer test, p is a Mersenne prime > 3 if and only if the smallest k such that p divides a(k) is k = (p - 1)/2.
For primes p, p^2 divides a(k) for some k if and only if p = 2 or p is in A238490. If p > 2, the only possible values for k are the divisors of (p - Legendre(3,p))/2 (e.g., 103^2 divides a(52) = 53028360515521 = 103^2 * 4998431569).
Conjecturally there must be infinitely many primes p such that a(p) is prime, but no such p is known. [By the formula below, there is no such p. - Jianing Song, Oct 31 2024]

			For n = 8 we have: a(1) = A001353(1), a(1)*a(2) = A001353(2), a(1)*a(2)*a(4) = A001353(4), a(1)*a(2)*a(4)*a(8) = A001353(8). The solution is a(1) = 1, a(2) = 4, a(4) = 14 and a(8) = 194.

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number
Wikipedia, Lucas Lehmer Primality Test

Cf. A001353, A238490, A309040.

Similar sequences: A061446, A008555.

b(n) = if(n==1, [1], my(v=vector(n)); v[1]=1; v[2]=4; for(i=3, n, v[i]=4*v[i-1]-v[i-2]); v)
a(n) = my(d=divisors(n)); prod(i=1, #d, (b(n)[d[i]])^moebius(n/d[i]))

Product_{d|n} a(d) = A001353(n), that is, a(n) = A001353(n)/(Product_{dA001353(d)^mu(n/d), where mu = A008683.

a(n) = A309040(2*n) for even n; A309040(n)*A309040(2*n) for odd n > 1. - Jianing Song, Oct 31 2024

A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

Original entry on oeis.org

1, 1, 4, 5, 17, 32, 97, 225, 628, 1573, 4225, 10880, 28769, 74849, 196708, 513765, 1347025, 3523360, 9229441, 24154625, 63251156, 165571781, 433507969, 1134881280, 2971250497, 7778684737, 20365103812, 53316141125, 139584105233, 365434903328, 956722661665

Offset: 1

Alexander Adamchuk, Aug 03 2006

Prime p divides a(p-1) for p={5,11,19,29,31,41,59,61,71,...} = A038872[n] Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. p^2 divides a(p-1) for prime p={11,19,29,31,41,59,61,71,...} = A045468[n] Primes congruent to {1, 4} mod 5. Square prime divisors of a(n) up to n=50 are{2,3,5,7,11,13,19,23,29,31,41,47,89,101,139,151,199,211,461,521,3571,9349}, It appears that square prime divisors of a(n) belong to A061446[n] Primitive part of Fibonacci(n), A001578[n] Smallest primitive prime factor of Fibonacci number F(n) and A072183[n] Sequence arising from factorization of the Fibonacci numbers. Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}] = A120297[n]. Sum[Sum[i+j-1,{i,1,n}],{j,1,n}] = n^3. Sum[Sum[(-1)^(i+j)*(i+j-1),{i,1,n}],{j,1,n}] = n for odd n and = 0 for even n.

			Matrix begins:
1 -1 2 -3 5
-1 2 -3 5 -8
2 -3 5 -8 13
-3 5 -8 13 -21
5 -8 13 -21 34

Colin Barker, Table of n, a(n) for n = 1..1000 [Terms up to n=200 from _Vincenzo Librandi_]
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,5,-1).

Cf. A120297, A000045, A038872, A001924, A062381, A038872, A045468, A061446, A001578, A072183.

Table[Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*fibonacci(i+j-1))) \\ Colin Barker, Mar 26 2015

Vec(-x*(x^3+2*x-1)/((x-1)*(x^2-3*x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Mar 26 2015

a(n) = Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}].
a(n) = 3*a(n-1)+a(n-2)-7*a(n-3)+5*a(n-4)-a(n-5) for n>5. - Colin Barker, Mar 26 2015
G.f.: -x*(x^3+2*x-1) / ((x-1)*(x^2-3*x+1)*(x^2-x-1)). - Colin Barker, Mar 26 2015

A126069 Generates A001350, the associated Mersenne numbers; A001350(n)=Product[a(d)] for d|n.

Original entry on oeis.org

1, 1, 4, 5, 11, 4, 29, 9, 19, 11, 199, 4, 521, 29, 31, 49, 3571, 19, 9349, 25, 211, 199, 64079, 36, 15251

Offset: 1

John W. Layman, Feb 28 2007

nonn

A 2001 Iranian Mathematical Olympiad question shows that such a generating sequence {a(n)} exists for the sequence {S(n)} whenever gcd(S(m),S(n)) = S(gcd(m,n)).

			The divisors of 6 are 1,2,3,6 and a(1)*a(2)*a(3)*a(6)=1*1*4*4=16, which is, in fact, A001350(6).

Cf. A001350, A061446.

A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.

Original entry on oeis.org

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 31, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 211, 433494437, 307, 109441, 461, 2971215073, 1103, 6168709, 151

Offset: 1

Jonathan Sondow, Oct 12 2015

nonn

Carmichael proved that a(n) > 1 if n > 12.

See A001578 (smallest primitive prime factor of F(n)) and A061446 (primitive part of F(n)) for more links.

			The prime factors of F(46)= 139 * 461 * 28657 that do not divide any smaller Fibonacci number are 139 and 461, so a(46) = 461.

R. D. Carmichael, On the numerical factors of the arithmetic forms α^n±β^n, Annals of Math., 15 (1913), 30-70.
Blair Kelly, Fibonacci and Lucas Factorizations
Ron Knott, Fibonacci numbers and special prime factors
Wikipedia, Carmichael's theorem

Cf. A000045, A001578, A061446, A086597, A152012, A152013.

prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, Last[p]], {n, 50}]

use ntheory ":all"; my %s; for (1..100) { my @f = factor(lucasu(1,-1,$)); pop @f while @f && $s{$f[-1]}++; say "$ ", $f[-1] || 1; }  # Dana Jacobsen, Oct 13 2015

A274584 Composite numbers n such that primitive part of Fibonacci(n) is prime.

Original entry on oeis.org

4, 8, 9, 10, 14, 15, 16, 18, 20, 21, 22, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 45, 51, 52, 54, 60, 62, 63, 65, 66, 72, 74, 75, 76, 82, 93, 94, 98, 105, 106, 111, 112, 119, 121, 122, 123, 124, 132, 135, 136, 140, 142, 144, 145, 152, 156, 158, 160, 172, 180

Offset: 1

Arkadiusz Wesolowski, Jun 29 2016

nonn

For every n > 1, A105602(a(n)) > 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..350
C. K. Caldwell, Top Twenty page, Fibonacci Primitive Part

Cf. A061446. Supersequence of A061254.

lst:=[]; for n in [4..180] do if not IsPrime(n) then d:=Divisors(n); p:=Truncate(&*[Fibonacci(d[i])^MoebiusMu(Truncate(n/d[i])): i in [1..#d]]); if IsPrime(p) then Append(~lst, n); end if; end if; end for; lst;

A309041 Irregular table read by rows: Let P(n,x) be the (monic) minimal polynomial of 2i*cos(Pi/n), where i = sqrt(-1) is the imaginary unit, then a(n,k) = [x^(2k)] P(n,x), n >= 3.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 6, 5, 1, 2, 4, 1, 1, 9, 6, 1, 5, 5, 1, 1, 15, 35, 28, 9, 1, 1, 4, 1, 1, 21, 70, 84, 45, 11, 1, 7, 14, 7, 1, 1, 24, 26, 9, 1, 2, 16, 20, 8, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 3, 9, 6, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1

Offset: 3

Jianing Song, Jul 08 2019

For n >= 3, it is easy to see that [x^(2k+1)] P(n,x) = 0, so they are omitted.
Row n (n >= 3) has length A023022(n) + 1 = phi(n)/2 + 1.
Let {U(n,x)} be defined as: U(0,x) = 0, U(1,x) = 1, U(n,x) = x*U(n-1,x) + U(n-2,x) for n >= 2, then U(n,x) = Product_{k|n, k>=2} P(k,x) for n > 0, because U(n,x) = Product_{m=1..n-1} (x - 2i*cos(Pi*m/n)) for n > 0.

			P(1,x) = x^2 + 4;
P(2,x) = x;
P(3,x) = x^2 + 1;
P(4,x) = x^2 + 2;
P(5,x) = x^4 + 3x^2 + 1;
P(6,x) = x^2 + 3;
P(7,x) = x^6 + 5x^4 + 6x^2 + 1;
P(8,x) = x^4 + 4x^2 + 2;
P(9,x) = x^6 + 6x^4 + 9x^2 + 1;
P(10,x) = x^4 + 5x^2 + 5;
...

Cf. A232624.
Cf. P(n,k): A061446 (k=1), A008555 (k=2), A253807 (k=3);
Cf. also A023022, A008683.

ro[n_] := (P = CoefficientList[p = MinimalPolynomial[2*I*Cos[Pi/n], x], x^2]; P); Flatten[Table[ro[n], {n, 3, 30}]]

U(n) = sum(i=0, (n-1)/2,binomial(n,2*i+1)*(poly/2)^(n-2*i-1)*((poly^2+4)/4)^i)
P(n) = if(n==1, poly^2+4, my(v=divisors(n)); prod(i=1, #v, U(n/v[i])^moebius(v[i])))
a(n,k) = polcoeff(P(n),2*k)

P(n,x) = Product_{0<=m<=n, gcd(m, n)=1} (x - 2i*cos(Pi*m/n)).
Equivalently, P(n,x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x^2 + 4*cos(Pi*m/n)) for n != 2. This shows that all terms are positive.
P(n,x) = Product_{k|n} U(n/k,x)^mu(k), mu = A008683.
Let MPR2(n,x) be the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624, then: for even n > 2, P(n,x) = MPR2(2n,i*x)*(-1)^A023022(n); for odd n, P(n,x) = MPR2(n,i*x)*MPR2(2n,i*x)*(-1)^A023022(n), i = sqrt(-1).
For n > 2, P(n,x) = MPR2(n,-x^2-2)*(-1)^A023022(n).
For n > 1, P(n,1) = A061446(n), P(n,2) = A008555(n), P(n,3) = A253807(n), ...
For even n > 2, a(n,k) = (-1)^(A023022(n)-k)*A232624(2n,2k).

cp's OEIS Frontend

A265578 LCM-transform of number of divisors function (A000005).

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A306825 Primitive part of A001353(n).

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A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

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A126069 Generates A001350, the associated Mersenne numbers; A001350(n)=Product[a(d)] for d|n.

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A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.

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A274584 Composite numbers n such that primitive part of Fibonacci(n) is prime.

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Magma

A309041 Irregular table read by rows: Let P(n,x) be the (monic) minimal polynomial of 2i*cos(Pi/n), where i = sqrt(-1) is the imaginary unit, then a(n,k) = [x^(2k)] P(n,x), n >= 3.

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