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A327420 Building sums recursively with the divisibility properties of their partial sums.

%I A327420 #36 Nov 20 2019 05:07:56
%S A327420 1,0,2,3,6,5,9,7,15,4,14,11,21,13,16,8,35,17,26,19,30,12,28,23,46,18,
%T A327420 38,10,49,29,45,31,77,20,50,27,63,37,52,24,68,41,54,43,74,25,64,47,96,
%U A327420 34,62,32,95,53,70,42,94,36,86,59,91,61,88,33,166,51,85
%N A327420 Building sums recursively with the divisibility properties of their partial sums.
%C A327420 Let R(n) = [k : n + 1 >= k >= 2] and divsign(s, k) = 0 if k does not divide s, else k if s/k is even and else -k. Compute s(k) = s(k+1) + divsign(s(k+1), k) with initial value s(n+2) = n + 1, k running down from n + 1 to 2. Then a(n) = s(2) if n > 0 and a(0) = s(n+2) = 0 + 1 = 1 as R(0) is empty in this case.
%C A327420 Examples: If n = 8 then R(8) = [9, 8, ..., 2] and the partial sums s are [0, 8, 8, 8, 8, 12, 15, 15] giving a(8) = 15. If p is prime, then the partial sums are [0, p, p, ..., p] since p is the only integer in R(p) diving p, i. e. the primes are the fixed points of this sequence. In the example section the computation of a(9) is traced.
%C A327420 Apparently the sequence is a permutation of the nonnegative integers.
%H A327420 Peter Luschny, <a href="/A327420/b327420.txt">Table of n, a(n) for n = 0..10000</a>
%H A327420 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the nonnegative integers</a>
%F A327420 For p prime, a(p) = p. - _Bernard Schott_, Sep 14 2019
%e A327420 The computation of a(9) = 4:
%e A327420 [ k: s(k) = s(k+1) + divsign(s(k+1),k)]
%e A327420 [10:   0,    10,       -10]
%e A327420 [ 9:   9,     0,         9]
%e A327420 [ 8:   9,     9,         0]
%e A327420 [ 7:   9,     9,         0]
%e A327420 [ 6:   9,     9,         0]
%e A327420 [ 5:   9,     9,         0]
%e A327420 [ 4:   9,     9,         0]
%e A327420 [ 3:   6,     9,        -3]
%e A327420 [ 2:   4,     6,        -2]
%p A327420 divsign := (s, k) -> `if`(irem(s, k) <> 0, 0, (-1)^iquo(s,k)*k):
%p A327420 A327420 := proc(n) local s, k; s := n + 1;
%p A327420     for k from s by -1 to 2 do
%p A327420         s := s + divsign(s, k) od;
%p A327420 return s end:
%p A327420 seq(A327420(n), n=0..66);
%o A327420 (SageMath)
%o A327420 def A327420(n):
%o A327420     s = n + 1
%o A327420     r = srange(s, 1, -1)
%o A327420     for k in r:
%o A327420         if k.divides(s):
%o A327420             s += (-1)^(s//k)*k
%o A327420     return s
%o A327420 print([A327420(n) for n in (0..66)])
%o A327420 (Julia)
%o A327420 divsign(s, k) = rem(s, k) == 0 ? (-1)^div(s, k)*k : 0
%o A327420 function A327420(n)
%o A327420     s = n + 1
%o A327420     for k in n+1:-1:2 s += divsign(s, k) end
%o A327420     s
%o A327420 end
%o A327420 [A327420(n) for n in 0:66] |> println
%Y A327420 Cf. A327093, A327487, A057032, A069829.
%K A327420 nonn
%O A327420 0,3
%A A327420 _Peter Luschny_, Sep 14 2019