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

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

Offset: 0

Author

Peter Luschny, Sep 14 2019

Keywords

• nonn

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.

Examples

			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]
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..10000
• Index entries for sequences that are permutations of the nonnegative integers

Crossrefs

Cf. A327093, A327487, A057032, A069829.

Programs

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

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);

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)])

Formula

For p prime, a(p) = p. - Bernard Schott, Sep 14 2019