cp's OEIS Frontend

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

%I A331870 #23 Mar 02 2020 08:53:02
%S A331870 2,758642,7057466,10805846,50860946,59677526,61800878,155045678,
%T A331870 178551374,217281146,343943882,359455694,432175586,609069506,
%U A331870 1449599486,1721358698,1829675354,1884592706,2013264194,2116706282,2680549946,2971193186,3084402122,3252387386,3454785386
%N A331870 Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24.
%C A331870 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.
%C A331870 a(2) was found by _Don Reble_, cf. A124456.
%C A331870 If we consider F(n+2) = 1 + the sum of the first n nonzero Fibonacci numbers (cf. A000071), then for even n we find:
%C A331870      4 divides F(n+2) for n == 4 (mod 12), 3 divides F(n+2) for n == 6 (mod 12),
%C A331870   F(n+2) == 3 (mod 4) for n == 8 (mod 12), 2 divides F(n+2) for n == 10 (mod 12),
%C A331870   F(n+2) == 5 (mod 6) for n == 12 (mod 24).
%C A331870   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).
%H A331870 Giovanni Resta, <a href="/A331870/b331870.txt">Table of n, a(n) for n = 1..80</a>
%F A331870 a(n) == 2 (mod 12) for all n.
%o A331870 (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.
%Y A331870 Cf. A124456, A331976, A111035, A000045 (Fibonacci numbers), A000071 (F(n)-1 = F(0)+...+F(n-2)).
%K A331870 nonn
%O A331870 1,1
%A A331870 _M. F. Hasler_, Feb 29 2020
%E A331870 Terms a(15) and beyond from _Giovanni Resta_, Mar 02 2020