A278833 -id:A278833 - cp's OEIS frontend

A278834 Record values of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

0, 2, 6, 8, 15, 30, 32, 40, 44, 46, 51, 57, 92

Offset: 1

M. F. Hasler, Dec 28 2016

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).

Conjectured to be finite and complete. Indeed it appears that 0.9*n < A004090(n) < n for all sufficiently large n.

Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278833 (the indices corresponding to the record values listed here), A264935.

Union@ Rest@ FoldList[Max, 0, #] &@ Table[Plus @@ IntegerDigits@ Fibonacci@ n - n, {n, 0, 10^4}] (* Michael De Vlieger, Dec 28 2016 *)

m=-1; for(k=0,1e4, sumdigits(fibonacci(k))-k>m && print1(m=sumdigits(fibonacci(k))-k,","))

A280185 a(n) = n - A004090(n), where A004090 is the sum of digits of the Fibonacci numbers A000045.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, -2, 3, 5, 2, 0, -6, 3, 5, -3, 8, -8, -5, -1, 5, -4, 1, 5, -5, -3, 6, 7, -2, 7, 6, 13, 0, 2, -1, -3, 0, 9, 2, -6, -4, 16, 10, -4, 2, 11, 16, 11, 10, -6, -6, 4, 22, 4, 12, 1, -3, 8, 5, -15, 15, 6, 8, 0, 2, 13, -2, -7, 8, 17, 4, 8, 25, 0, 9, -8, 10, 10, -9, -2, 21, -4, 2, 18, -15, 12, -4, 6, -10, 19, -5, 17, 23, 14, 28, 5, 4, 6, -3, 16, -2

Offset: 0

M. F. Hasler, Dec 28 2016

Conjectured to increase to infinity. It appears that the slope of A004090(n) is roughly 0.93, at least in the range 0..10^5.

I conjecture that this sequence takes its minimum at a(2619) = -92. - M. F. Hasler, Dec 30 2016

			a(2) = a(3) = 1 because A000045(2) = 1 = A004090(2), its digital sum, A000045(3) = 2 = A004090(3), and 2 - 1 = 3 - 2 = 1.

M. F. Hasler, Table of n, a(n) for n = 0..100000

Cf. A000045 (Fibonacci numbers), A004090 (their digital sums), A278834 & A278833 (record values of this sequence and corresponding indices), A264935.

Table[n - Total@ IntegerDigits@ Fibonacci@ n, {n, 0, 99}] (* Michael De Vlieger, Dec 28 2016 *)

A280185(n)=n-sumdigits(fibonacci(n))
/* To produce the b-file; can be used for searches or similar purpose, this is faster than to compute fib(n) anew for each term. */
b=-a=1;for(n=0,1e5,write("/tmp/A280185.txt",n" ",n-sumdigits(a=b+b=a)))

a(n) = n - A004090(n) = n - A007953(A000045(n)).

cp's OEIS Frontend

A278834 Record values of A004090(n) - n, where A004090 is the sum of digits of the Fibonacci numbers A000045.

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