A336095 - OEIS

A336095

a(n) = a(f(n)) + a(n-f(n)) with a(1) = a(2) = 1 (f = A000006).

1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 27, 27, 28, 29, 30, 31, 31, 32, 33, 33, 33, 34, 34, 35, 36, 36, 37, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 44, 44, 44

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OFFSET

1,3

COMMENTS

If Legendre's conjecture is true, then this sequence hits every positive integer.

Does the lim_{n->infinity} a(n)/n exist? If it exists, what is its value?

LINKS

Table of n, a(n) for n=1..75.

Wikipedia, Legendre's conjecture

MATHEMATICA

f[n_] := IntegerPart[Sqrt[Prime[n]]]; a[1] = a[2] = 1; a[n_] := a[n] = a[(f1 = f[n])] + a[n - f1]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)

PROG