A000032 -id:A000032 - cp's OEIS frontend

A191924 Ordered sums 2f+3g, where f and g are Lucas numbers (A000032 beginning at 1).

5, 9, 11, 14, 15, 17, 18, 20, 23, 25, 26, 27, 29, 31, 34, 35, 39, 41, 43, 45, 47, 48, 55, 56, 57, 60, 61, 62, 67, 68, 69, 70, 76, 79, 89, 90, 91, 93, 95, 97, 101, 103, 106, 109, 112, 115, 123, 127, 143, 145, 147, 148, 149, 155, 161, 163, 164, 173, 177, 181

Offset: 1

Clark Kimberling, Jun 19 2011

nonn

Cf. A191925, A191926, A191927.

c = 2; d = 3; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A191924 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A191925:c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A191926:d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A191927 *)

A191976 Ordered sums 3f+4g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

7, 13, 15, 16, 19, 21, 24, 25, 28, 31, 33, 37, 40, 45, 47, 49, 53, 56, 58, 61, 65, 66, 70, 75, 77, 81, 82, 84, 91, 93, 98, 99, 103, 105, 115, 119, 125, 126, 128, 131, 137, 145, 149, 153, 157, 159, 169, 170, 185, 191, 197, 200, 203, 209, 213, 221, 232, 240

Offset: 1

Clark Kimberling, Jun 20 2011

nonn

Cf. A191977, A191978, A191979, A191929.

c = 3; d = 4; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A191976 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A191977: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A191978: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A191979 *)

A191980 Ordered sums f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

6, 8, 9, 12, 16, 18, 19, 21, 22, 23, 24, 26, 27, 31, 33, 34, 36, 38, 39, 42, 44, 46, 49, 52, 53, 56, 58, 59, 62, 64, 66, 67, 73, 81, 82, 84, 91, 93, 94, 96, 97, 101, 102, 108, 111, 119, 128, 131, 137, 138, 143, 146, 148, 149, 152, 156, 158, 163, 166, 174

Offset: 1

Clark Kimberling, Jun 20 2011

nonn

Cf. A191981, A191982, A191983.

c = 1; d = 5; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A191980 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A191981: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A191982: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A191983 *)
With[{nn=20},Select[Union[(#[[1]]+5#[[2]]&/@Tuples[LucasL[Range[nn]],2])],#<=LucasL[nn]+5&]] (* Harvey P. Dale, Dec 07 2018 *)

A192045 Ordered sums 3f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

8, 14, 17, 18, 23, 24, 26, 27, 29, 32, 36, 38, 41, 44, 47, 48, 53, 56, 58, 59, 64, 67, 68, 69, 74, 76, 88, 89, 92, 93, 99, 102, 107, 109, 111, 122, 123, 142, 144, 146, 148, 154, 156, 157, 161, 166, 176, 177, 178, 196, 199, 231, 232, 233, 238, 243, 244, 247

Offset: 1

Clark Kimberling, Jun 21 2011

nonn

Cf. A000204, A191988, A192046, A192047, A192048.

c = 3; d = 5; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A192045 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A192046: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A192047: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A192048 *)
Union[3First[#]+5Last[#]&/@Tuples[LucasL[Range[10]],2]] (* Harvey P. Dale, May 04 2012 *)

A192049 Ordered sums 4f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

Original entry on oeis.org

9, 17, 19, 21, 24, 27, 31, 32, 33, 36, 39, 43, 47, 48, 49, 51, 59, 63, 64, 67, 71, 77, 79, 83, 87, 92, 94, 99, 102, 106, 107, 118, 121, 127, 131, 134, 136, 149, 151, 157, 161, 162, 171, 173, 189, 193, 203, 206, 208, 217, 223, 239, 243, 247, 251, 261, 263

Offset: 1

Clark Kimberling, Jun 21 2011

nonn

Cf. A192050, A192051, A192052.

c = 4; d = 5; f[n_] := LucasL[n];
g[n_] := c*f[n]; h[n_] := d*f[n];
t[i_, j_] := h[i] + g[j];
u = Table[t[i, j], {i, 1, 20}, {j, 1, 20}];
v = Union[Flatten[u]]    (* A192049 *)
t1[i_, j_] := If[g[i] - h[j] > 0, g[i] - h[j], 0]
u1 = Table[t1[i, j], {i, 1, 20}, {j, 1, 20}];
v1 = Union[Flatten[u1]]  (* A192050: c*f(i)-d*f(j) *)
g1[n_] := d*f[n]; h1[n_] := c*f[n];
t2[i_, j_] := If[g1[i] - h1[j] > 0, g1[i] - h1[j], 0]
u2 = Table[t2[i, j], {i, 1, 20}, {j, 1, 20}];
v2 = Union[Flatten[u2]]  (* A192051: d*f(i)-c*f(j) *)
v3 = Union[v1, v2]       (* A192052 *)

A203579 Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.

Original entry on oeis.org

2, 2, 7, 17, 57, 177, 577, 1857, 6017, 19457, 62977, 203777, 659457, 2134017, 6905857, 22347777, 72318977, 234029057, 757334017, 2450784257, 7930904577, 25664946177, 83053510657, 268766806017, 869747654657, 2814562533377, 9108115685377, 29474481504257

Offset: 0

Wolfdieter Lang, Jan 14 2012

			With A000032 = {2,1,3,4,7,...},
  2*a(4) = 1*2*7 + 4*1*4 + 6*3*3 + 4*4*1 + 1*7*2 = 114.

Michael De Vlieger, Table of n, a(n) for n = 0..1961
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.

Cf. A000032, A014335.

Array[Sum[Binomial[#, k] LucasL[k] LucasL[# - k], {k, 0, #}]/2 &, 28, 0] (* Michael De Vlieger, Dec 28 2020 *)

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n)/2, n>=0, with L(n)=A000032(n).
E.g.f.: (1/2)*(exp(phi*x)+exp(-(phi-1)*x))^2 =
  exp(x)*(cosh(sqrt(5)*x)+1), with the golden section phi:=(1+sqrt(5))/2. (See the e.g.f. of A000032).
a(n) = 2^(n-1)*L(n) + 1.
a(n) = 5*A014335(n) + 2. - Vladimir Reshetnikov, Oct 06 2016

A223486 Lucas entry points: a(n) = least k such that n divides Lucas number L_k (=A000032(k), for k >= 0), or -1 if there is no such k.

Original entry on oeis.org

0, 0, 2, 3, -1, 6, 4, -1, 6, -1, 5, -1, -1, 12, -1, -1, -1, 6, 9, -1, -1, 15, 12, -1, -1, -1, 18, -1, 7, -1, 15, -1, -1, -1, -1, -1, -1, 9, -1, -1, 10, -1, 22, 15, -1, 12, 8, -1, 28, -1, -1, -1, -1, 18, -1, -1, -1, 21, 29, -1, -1, 15, -1, -1, -1, -1, 34, -1

Offset: 1

Casey Mongoven, Mar 20 2013

sign

If one takes L_k, for k >= 1, that is A000204, then a(1) = 1 and a(2) = 3 followed by the given numbers. This fits then with A106291(n) = A253808(n)*a(n), n >= 1 (where in A253808 a negative entry at position n indicates, as in the present sequence, that the Lucas numbers are not divisible by n. For odd primes not dividing any Lucas numbers see A053028. No power 2^m, m >= 3 divides any Lucas number, see, e.g., Vajda, p. 81). - Wolfdieter Lang, Jan 20 2015

			a(9) = 6 because L_6 = 18 is the first number in the Lucas sequence (A000032) that 9 divides.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 25.
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000032, A000204, A001177, A194363, A053028 (primes not dividing any Lucas numbers), A106291, A253808.

test[n_] := Module[{a, b, t, cnt = 1}, {a, b} = {2, 1}; While[cnt++; t = b; b = Mod[a + b, n]; a = t; ! (b == 0 || {a, b} == {2, 1})]; If[b == 0, cnt, -1]]; Join[{0, 0}, Table[test[i], {i, Range[3, 100]}]] (* T. D. Noe, Mar 22 2013 *)

Edited. Added "k >= 0" in the name and added cross references. - Wolfdieter Lang, Jan 20 2015

A256178 Expansion of exp( Sum_{n >= 1} L(2n)L(4n)x^n/n ), where L(n) = A000032(n) is a Lucas number.

Original entry on oeis.org

1, 21, 385, 6930, 124410, 2232594, 40062659, 718896255, 12900072515, 231482415780, 4153783429236, 74536619356836, 1337505365115205, 24000559953034665, 430672573790340805, 7728105768275278134, 138675231255170368494

Offset: 0

Peter Bala, Mar 18 2015

Let L(n) = A000032(n) denote the n-th Lucas number.
For a fixed positive integer k, the power series expansion of exp( Sum_{n >= 1} L(k*n)x^n/n ) has integer coefficients given by the formula F(k*n)/F(k), where F(n) = A000045(n) [Johnson, 2.22].
The power series expansion of exp( Sum_{n >= 1} L(k*n)*L(2*k*n) *x^n/n ) has integer coefficients given by ( F(k*(n + 1))*F(k*(n + 2))*F(k*(n + 3)) )/( F(k)*F(2*k)*F(3*k) )
The present sequence is the particular case k = 2. See A001655 for the case k = 1.

B. Johnson, Fibonacci Identities by Matrix Methods and Generalisation to Related Sequences
Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (21,-56,21,-1).

Cf. A000032, A000045, A001655, A010048, A153386, A265288.

seq((1/24)*fibonacci(2*n+2)*fibonacci(2*n+4)*fibonacci(2*n+6), n = 0 .. 16);

Table[1/8 * Sum[Fibonacci[2*k + 2]*Fibonacci[6*n - 6*k + 6], {k, 0, n}], {n, 0, 17}] (* or *) RecurrenceTable[{a[n] == 21*a[n - 1] - 56*a[n - 2] + 21*a[n - 3] - a[n - 4], a[1] == 1, a[2] == 21, a[3] == 385, a[4] == 6930}, a, {n, 17}] (* Michael De Vlieger, Mar 18 2015 *)

a(n) = ( F(2*n + 2)*F(2*n + 4)*F(2*n + 6) )/( F(2)*F(4)*F(6) ).
a(n) = (1/8) * Sum_{k = 0..n} F(2*k + 2)*F(6*n - 6*k + 6).
O.g.f.: 1/( (1 - 3*x + x^2)*(1 - 18*x + x^2) ) = 1/8 * Sum_{n >= 0} F(2*n + 2)*x^n * Sum_{n >= 0} F(6*n + 6)*x^n.
O.g.f. also equals exp( Sum_{n >= 1} trace( M^(2*n) + M^(6*n) )*x^n/n ), where M is the 2X2 matrix [ 1, 1; 1, 0 ].
Recurrences: a(n) = 21*a(n-1) - 56*a(n-2) + 21*a(n-3) - a(n-4).
Also a(0) = 1 and for n >= 1, a(n) = (1/n)*Sum_{k = 1..n} L(2*k)*L(4*k)*a(n-k).
From Peter Bala, Aug 19 2022: (Start)
Sum_{n >= 0} 1/a(n) = 40/3 - 8*Sum_{n >= 1} 1/F(2*n) = 40/3 - 8*A153386.
Sum_{n >= 0} (-1)^n/a(n) = - 88/3 + 40*Sum_{n >= 1} (-1)^(n+1)/F(2*n). Cf. A265288. (End)

A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

Original entry on oeis.org

2, 1, 3, 2, 7, 11, 2, 29, 47, 2, 3, 199, 2, 521, 3, 2, 2207, 3571, 2, 9349, 7, 2, 3, 139, 2, 11, 3, 2, 7, 59, 2, 3010349, 1087, 2, 3, 11, 2, 54018521, 3, 2, 47, 370248451, 2, 6709, 7, 2, 3, 6643838879, 2, 29, 3, 2, 7, 119218851371, 2, 11, 47, 2, 3, 709, 2

Offset: 0

Vincenzo Librandi, Dec 26 2016

nonn

From Robert Israel, Jan 05 2017: (Start)
If m and n are odd, m > 1 and m | n, then a(n) <= a(m).
a(n) = 2 if and only if 3 | n.
a(n) = 3 if and only if n is in A091999.
a(n) is never 5.
a(n) = 7 if and only if n is in A259755.
a(n) = A000032(n) if and only if n is in A001606.
(End)

Robert Israel, Table of n, a(n) for n = 0..1000
Fibonacci and Lucas Factorizations

Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.

Column k=2 of A238899 (for n>=2).

[2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];

lucas:= n -> combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1):
spf:= proc(n) local F;
  F:= remove(hastype,ifactors(n,easy)[2],symbol);
  if F <> [] then return min(seq(f[1],f=F)) fi;
min(numtheory:-factorsec(n))
end proc:
spf(1):= 1:
map(spf @ lucas, [$0..200]); # Robert Israel, Jan 05 2017

f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]

a000032(n) = fibonacci(n+1)+fibonacci(n-1)
a(n) = if(a000032(n-1)==1, 1, factor(a000032(n-1))[1, 1]) \\ Felix Fröhlich, Dec 26 2016

a(n) = A020639(A000032(n)). - Felix Fröhlich, Dec 26 2016

Offset changed from Bruno Berselli, Dec 27 2016

A304091 a(n) is the number of the proper divisors of n that are Lucas numbers (A000032, with 2 included).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 1, 3, 1, 3, 3, 3, 1, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 2, 5, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 5, 1, 2, 2, 3, 3, 3, 1, 3, 2, 2, 1, 5, 1, 2, 3, 4, 1, 4, 2, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000032, A093540, A102460, A304092, A304093.

Cf. also A293435, A300836.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304091(n) = sumdiv(n,d,(dA102460(d));

a(n) = Sum_{d|n, dA102460(d).
a(n) = A304092(n) - A102460(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2 + A093540 = 2.462858... . - Amiram Eldar, Jul 05 2025

cp's OEIS Frontend

A191924 Ordered sums 2*f+3*g, where f and g are Lucas numbers (A000032 beginning at 1).

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A191976 Ordered sums 3*f+4*g, where f and g are Lucas numbers (A000032 beginning at 1).

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A191980 Ordered sums f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

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A192045 Ordered sums 3*f+5*g, where f and g are Lucas numbers (A000032 beginning at 1).

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A192049 Ordered sums 4*f+5*g, where f and g are Lucas numbers (A000032 beginning at 1).

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A203579 Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.

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A223486 Lucas entry points: a(n) = least k such that n divides Lucas number L_k (=A000032(k), for k >= 0), or -1 if there is no such k.

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A256178 Expansion of exp( Sum_{n >= 1} L(2*n)*L(4*n)*x^n/n ), where L(n) = A000032(n) is a Lucas number.

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A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

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A191924 Ordered sums 2f+3g, where f and g are Lucas numbers (A000032 beginning at 1).

A191976 Ordered sums 3f+4g, where f and g are Lucas numbers (A000032 beginning at 1).

A192045 Ordered sums 3f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

A192049 Ordered sums 4f+5g, where f and g are Lucas numbers (A000032 beginning at 1).

A256178 Expansion of exp( Sum_{n >= 1} L(2n)L(4n)x^n/n ), where L(n) = A000032(n) is a Lucas number.