A331870 Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24.

2, 758642, 7057466, 10805846, 50860946, 59677526, 61800878, 155045678, 178551374, 217281146, 343943882, 359455694, 432175586, 609069506, 1449599486, 1721358698, 1829675354, 1884592706, 2013264194, 2116706282, 2680549946, 2971193186, 3084402122, 3252387386, 3454785386

Offset: 1

Author

M. F. Hasler, Feb 29 2020

Keywords

• nonn

Comments

A111035 lists numbers n which divide the sum of the first n nonzero Fibonacci numbers. Most of these are multiples of 24. Sequence A124456 lists those which aren't. Most of these are odd (cf. A331976), this sequence lists the exceptions.

a(2) was found by Don Reble, cf. A124456.

If we consider F(n+2) = 1 + the sum of the first n nonzero Fibonacci numbers (cf. A000071), then for even n we find:

4 divides F(n+2) for n == 4 (mod 12), 3 divides F(n+2) for n == 6 (mod 12),

F(n+2) == 3 (mod 4) for n == 8 (mod 12), 2 divides F(n+2) for n == 10 (mod 12),

F(n+2) == 5 (mod 6) for n == 12 (mod 24).

These relations imply that all terms a(n) == 2 (mod 12) for all n. This also means that all terms of A111035 are either divisible by 24, or odd, or congruent to 2 (mod 12).

Links

• Giovanni Resta, Table of n, a(n) for n = 1..80

Crossrefs

Cf. A124456, A331976, A111035, A000045 (Fibonacci numbers), A000071 (F(n)-1 = F(0)+...+F(n-2)).

Programs

PARI

M=[1,1;1,0]; forstep(n=2,oo,12,n%24&&(Mod(M,n)^(n+1))[1,1]==1&& print1(n",")) \\ Custom implementation of is_A111035(), check for updates there.

Formula

a(n) == 2 (mod 12) for all n.

Extensions

Terms a(15) and beyond from Giovanni Resta, Mar 02 2020