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a(n+1) is the smallest divisor of (2 + sum of first n terms) but not occurring earlier; a(1) = 1.
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%I #33 Jan 26 2022 02:30:50

%S 1,3,2,4,6,9,27,18,8,5,17,34,68,12,24,10,25,11,13,23,7,47,94,235,15,

%T 16,32,48,51,289,578,102,36,26,73,219,30,20,14,46,50,470,60,40,146,21,

%U 49,28,113,29,19,35,42,54,64,22,77,329,84,56,292,365,65,37,131,38,33

%N a(n+1) is the smallest divisor of (2 + sum of first n terms) but not occurring earlier; a(1) = 1.

%C Conjecture: integer permutation with inverse A095259: a(A095259(n))=A095259(a(n))=n. - Comment revised: _Reinhard Zumkeller_, Dec 31 2014

%C A095260(n) = a(a(n)).

%C First fixed points: 1, 4, 54, 416, ...

%C A253415(n) = smallest missing number within the first n terms. - _Reinhard Zumkeller_, Dec 31 2014

%H Rémy Sigrist, <a href="/A095258/b095258.txt">Table of n, a(n) for n = 1..10000</a> (first 797 terms from Reinhard Zumkeller)

%H Michael De Vlieger, <a href="/A095258/a095258.png">Log-log scatterplot of a(n)</a> for n = 1..2^16.

%H Michael De Vlieger, <a href="/A095258/a095258_1.png">Annotated log-log scatterplot of a(n)</a> for n = 1..2^14 showing records in red, local minima in blue, primes in magenta, prime powers not prime in gold.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t c[_] = 0; j = c[1] = 1; s = 3; {j}~Join~Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; Sow[k]; j = k; s += k, {i, 2, 80}]][[-1, -1]] (* _Michael De Vlieger_, Jan 23 2022 *)

%o (Haskell)

%o import Data.List (delete)

%o a095258 n = a095258_list !! (n-1)

%o a095258_list = 1 : f [2..] 1 where

%o f xs z = g xs where

%o g (y:ys) = if mod z' y > 0 then g ys else y : f (delete y xs) (z + y)

%o z' = z + 2

%o -- _Reinhard Zumkeller_, Dec 31 2014

%o (Python)

%o from itertools import islice

%o from sympy import divisors

%o def A095258_gen(): # generator of terms

%o bset, s = {1}, 3

%o yield 1

%o while True:

%o for d in divisors(s):

%o if d not in bset:

%o yield d

%o bset.add(d)

%o s += d

%o break

%o A095258_list = list(islice(A095258_gen(),30)) # _Chai Wah Wu_, Jan 25 2022

%Y Cf. A253415.

%K nonn,look

%O 1,2

%A _Reinhard Zumkeller_, May 31 2004