A093540 -id:A093540 - cp's OEIS frontend

A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000204, A093540, A304091, A304094, A304095.

Cf. also A293435, A300836.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304093(n) = sumdiv(n,d,(dA000204(d));

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540. - Amiram Eldar, Jul 05 2025

A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 4, 1, 1, 2, 2, 1, 3, 1, 3, 2, 1, 2, 3, 2, 1, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 3, 3, 2, 1, 2, 2, 1, 1, 4, 1, 1, 3, 3, 1, 3, 2, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000204, A093540, A304092, A304093, A304096.

Cf. also A005086, A300837.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304094(n) = sumdiv(n,d,isA000204(d));

a(n) = A304092(n) - A059841(n).
a(n) = A304096(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 = 1.962858... . - Amiram Eldar, Dec 31 2023

A371649 Decimal expansion of Sum_{k>=0} 1/Lucas(5^k).

Original entry on oeis.org

1, 0, 9, 0, 9, 1, 5, 0, 5, 1, 7, 7, 0, 0, 7, 7, 6, 7, 0, 0, 1, 8, 6, 5, 7, 5, 0, 2, 4, 1, 4, 2, 2, 8, 2, 0, 5, 7, 1, 5, 1, 7, 5, 1, 0, 2, 3, 1, 9, 9, 0, 6, 6, 9, 8, 9, 0, 5, 0, 3, 2, 1, 7, 0, 9, 2, 2, 2, 4, 3, 0, 8, 1, 7, 5, 8, 2, 8, 8, 4, 4, 6, 4, 9, 0, 2, 6, 3, 1, 9, 6, 7, 3, 7, 2, 4, 8, 1, 8, 3, 1, 2, 4, 1, 7

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2001).

			1.09091505177007767001865750241422820571517510231990...

M. A. Nyblom, A Theorem on Transcendence of Infinite Series II, Journal of Number Theory, Vol. 91, No. 1 (2001), pp. 71-80.
Index entries for transcendental numbers.

Cf. A000032, A144837, A371650.

Similar constants: A093540, A338304, A338612, A371647.

RealDigits[Sum[1/LucasL[5^k], {k, 0, 10}], 10, 120][[1]]

suminf(k = 0, 1/(fibonacci(5^k-1) + fibonacci(5^k+1)))

Equals Sum_{k>=0} 1/A144837(k).

A243991 Decimal expansion of sum of reciprocals of Bell numbers for n>0.

Original entry on oeis.org

1, 7, 9, 2, 2, 6, 3, 0, 0, 4, 8, 5, 8, 5, 6, 3, 3, 0, 6, 0, 2, 6, 0, 8, 4, 3, 3, 9, 6, 7, 7, 3, 4, 9, 6, 2, 7, 2, 9, 8, 4, 7, 0, 8, 3, 7, 4, 7, 6, 1, 4, 9, 9, 6, 0, 4, 1, 0, 0, 4, 7, 9, 7, 9, 1, 6, 7, 2, 8, 3, 7, 4, 8, 3, 8, 1, 8, 7, 2, 4, 9, 8, 7, 3, 6, 9, 3, 6, 7, 8, 9, 0, 3, 8, 7, 1, 2, 4, 7, 4, 7, 2

Offset: 1

Franklin T. Adams-Watters, Jun 17 2014

1.792263004858...

Cf. A000110, A078506, A079586, A093540.

with(combinat, bell): evalf(sum(1/bell(n), n = 1..infinity), 120); # Vaclav Kotesovec, Jan 30 2015

A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

8, 3, 0, 5, 0, 2, 8, 2, 1, 5, 8, 6, 8, 7, 6, 6, 8, 2, 3, 1, 6, 9, 3, 6, 4, 8, 6, 2, 5, 1, 0, 5, 9, 5, 1, 9, 1, 7, 7, 3, 0, 4, 6, 2, 1, 4, 3, 0, 4, 0, 8, 2, 8, 0, 1, 4, 6, 0, 2, 6, 4, 1, 3, 9, 0, 7, 9, 1, 0, 4, 9, 8, 4, 8, 6, 0, 4, 3, 0, 0, 6, 7, 4, 9, 3, 3, 0

Offset: 0

Amiram Eldar, Nov 03 2020

André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)).

			0.83050282158687668231693648625105951917730462143040...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
Eric Weisstein's World of Mathematics, Reciprocal Lucas Constant.

Cf. A000032, A000045, A001622, A079586, A093540, A153415, A153416, A158933.

RealDigits[Sum[(-1)^(n+1)/LucasL[n], {n, 1, 1000}], 10, 120][[1]]

Equals A153416 - A153415.
Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/Fibonacci(2*k).
Equals Sum_{k>=1} (-1)^(k+1)/(phi^k + (1-phi)^k), where phi is the golden ratio (A001622).
Equals Sum_{k>=0} 1/(phi^(2*k+1) + (-1)^k).

A346588 Decimal expansion of the sum of reciprocals of tribonacci numbers A000213.

Original entry on oeis.org

3, 7, 7, 3, 9, 4, 8, 0, 6, 0, 1, 9, 7, 0, 1, 5, 8, 1, 8, 3, 8, 5, 4, 0, 2, 4, 2, 6, 6, 2, 9, 5, 1, 2, 7, 4, 9, 7, 6, 8, 0, 7, 4, 1, 7, 3, 2, 2, 2, 5, 8, 4, 3, 8, 0, 8, 8, 1, 3, 1, 6, 1, 8, 5, 0, 8, 4, 3, 3, 7, 8, 3, 8, 1, 7, 1, 7, 8, 1, 7, 2, 6, 3, 6, 5, 0, 4, 1, 2, 2, 5, 5, 8, 7, 9, 7, 4, 2, 3, 4, 5, 7, 5, 0, 1

Offset: 1

Christoph B. Kassir, Jul 24 2021

			3.7739480601970158183854024266295127497680741732225...

Cf. A000213, A079586, A084119, A093540, A153386, A153387.

RealDigits[Total[1/LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 500]], 10, 105][[1]] (* Amiram Eldar, Jul 26 2021 *)

More terms from Jon E. Schoenfield, Jul 25 2021

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A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

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A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

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A371649 Decimal expansion of Sum_{k>=0} 1/Lucas(5^k).

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A243991 Decimal expansion of sum of reciprocals of Bell numbers for n>0.

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A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).

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A346588 Decimal expansion of the sum of reciprocals of tribonacci numbers A000213.

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