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 A327420 Building sums recursively with the divisibility properties of their partial sums. 5
 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, 38, 10, 49, 29, 45, 31, 77, 20, 50, 27, 63, 37, 52, 24, 68, 41, 54, 43, 74, 25, 64, 47, 96, 34, 62, 32, 95, 53, 70, 42, 94, 36, 86, 59, 91, 61, 88, 33, 166, 51, 85 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. 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. Apparently the sequence is a permutation of the nonnegative integers. LINKS Peter Luschny, Table of n, a(n) for n = 0..10000 FORMULA For p prime, a(p) = p. - Bernard Schott, Sep 14 2019 EXAMPLE The computation of a(9) = 4: [ k: s(k) = s(k+1) + divsign(s(k+1),k)] [10:   0,    10,       -10] [ 9:   9,     0,         9] [ 8:   9,     9,         0] [ 7:   9,     9,         0] [ 6:   9,     9,         0] [ 5:   9,     9,         0] [ 4:   9,     9,         0] [ 3:   6,     9,        -3] [ 2:   4,     6,        -2] MAPLE divsign := (s, k) -> `if`(irem(s, k) <> 0, 0, (-1)^iquo(s, k)*k): A327420 := proc(n) local s, k; s := n + 1;     for k from s by -1 to 2 do         s := s + divsign(s, k) od; return s end: seq(A327420(n), n=0..66); PROG (SageMath) def A327420(n):     s = n + 1     r = srange(s, 1, -1)     for k in r:         if k.divides(s):             s += (-1)^(s//k)*k     return s print([A327420(n) for n in (0..66)]) (Julia) divsign(s, k) = rem(s, k) == 0 ? (-1)^div(s, k)*k : 0 function A327420(n)     s = n + 1     for k in n+1:-1:2 s += divsign(s, k) end     s end [A327420(n) for n in 0:66] |> println CROSSREFS Cf. A327093, A327487, A057032, A069829. Sequence in context: A035493 A070038 A328638 * A119790 A272214 A319785 Adjacent sequences:  A327417 A327418 A327419 * A327421 A327422 A327423 KEYWORD nonn AUTHOR Peter Luschny, Sep 14 2019 STATUS approved

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Last modified January 27 22:46 EST 2022. Contains 350654 sequences. (Running on oeis4.)