login
A331870
Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24.
3
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
1,1
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
FORMULA
a(n) == 2 (mod 12) for all n.
PROG
(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.
CROSSREFS
Cf. A124456, A331976, A111035, A000045 (Fibonacci numbers), A000071 (F(n)-1 = F(0)+...+F(n-2)).
Sequence in context: A237191 A367823 A101118 * A371642 A162927 A173363
KEYWORD
nonn
AUTHOR
M. F. Hasler, Feb 29 2020
EXTENSIONS
Terms a(15) and beyond from Giovanni Resta, Mar 02 2020
STATUS
approved