A248593 Least positive integer m such that m + n divides F(m), where F(m) is the m-th Fibonacci number given by A000045.

10, 6, 84, 12, 16, 7, 27, 9, 144, 30, 28, 12, 8, 30, 14, 18, 57, 19, 342, 18, 20, 24, 66, 12, 9, 27, 144, 60, 112, 35, 16, 24, 60, 55, 20, 12, 40, 111, 24, 36, 88, 72, 80, 48, 10, 15, 72, 24, 224, 18, 50, 54, 270, 72, 54, 33, 224, 18, 28, 12

Offset

1,1

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1) except for n = 1, 2, 3, 9.

In contrast, it is easy to show that for any integer n > 0, there is a positive integer m such that m + n divides 2^m - 1.

a(n) exists for any n > 0. See Bloom (1998). - Amiram Eldar, Jan 15 2022

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

David M. Bloom, Offset Entries, Solution to Problem B-830, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 89-90.

Example

a(1) = 10 since 10 + 1 = 11 divides F(10) = 55.

Mathematica

Do[m=1; Label[aa]; If[Mod[Fibonacci[m], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000045, A247937, A247940, A248588, A248590.

Sequence in context: A344103 A181107 A234974 * A038308 A185221 A349845

Adjacent sequences: A248590 A248591 A248592 * A248594 A248595 A248596

Keyword

nonn

Author

Zhi-Wei Sun, Oct 09 2014

Status

approved