A366912 Partial sums of A366911: a(1) = 0, and for n > 0, a(n+1) = a(n) + A366911(n).

0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 8, 5, 7, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 6, 7, 8, 9, 8, 9, 8

Offset: 1

Author

Rémy Sigrist, Oct 27 2023

Keywords

• nonn

Comments

By analogy with A064289, a(n) corresponds to the height of A364054(n) = number of addition steps - number of subtraction steps to produce it.

Examples

			a(5) = A366911(1) + A366911(2) + A366911(3) + A366911(4) = 1 + 1 + 1 - 1 = 2.
		

Links

• Rémy Sigrist, Colored scatterplot of the first 100000 terms of A364054 (where the color is function of a(n))
• Rémy Sigrist, PARI program

Crossrefs

Cf. A064289, A364054, A366911, A366913.

Programs

Mathematica

nn = 2^16; c[] := False; m[] := 0; j = 1; s = b[1] = 0;
  c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++]; s += (k - j)/p;
    Set[{a[n - 1], b[n - 1], c[k], j}, {(k - j)/p, s, True, k}],
    {n, 2, nn + 1}], n];
Array[b, nn] (* Michael De Vlieger, Oct 27 2023 *)

PARI

See Links section.

Python

from itertools import count, islice
from sympy import nextprime
def A366912_gen(): # generator of terms
    a, aset, p, c = 1, {0,1}, 2, 0
    while True:
        k, b = divmod(a,p)
        for i in count(-k):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                yield c
                c += i
                break
A366912_list = list(islice(A366912_gen(),30)) # Chai Wah Wu, Oct 27 2023

Formula

a(n) = Sum_{k = 1..n-1} A366911(k).