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%I #10 Apr 28 2018 03:14:18
%S 0,2,6,8,15,30,32,40,44,46,51,57,92
%N Record values of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.
%C For indices n = 1, 5, 83, 156 and 512, the value of the difference A004090(n) - n is the same as for the preceding record, namely (0, 0, 15, 15, 40), respectively. So the sequence of records in the weak sense of >= would be (0, 0, 0, 2, 6, 8, 15, 15, 15, 30, 32, 40, 40, 44, 46, 51, 57, 92).
%C Conjectured to be finite and complete. Indeed it appears that 0.9*n < A004090(n) < n for all sufficiently large n.
%t Union@ Rest@ FoldList[Max, 0, #] &@ Table[Plus @@ IntegerDigits@ Fibonacci@ n - n, {n, 0, 10^4}] (* _Michael De Vlieger_, Dec 28 2016 *)
%o (PARI) m=-1; for(k=0,1e4, sumdigits(fibonacci(k))-k>m && print1(m=sumdigits(fibonacci(k))-k,","))
%Y Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278833 (the indices corresponding to the record values listed here), A264935.
%K nonn,base,fini,full
%O 1,2
%A _M. F. Hasler_, Dec 28 2016
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