A377388 -id:A377388 - cp's OEIS frontend

A377389 Partial sums of A377388.

0, 1, -1, -4, -9, -17, -23, -34, -47, -68, -84, -75, -117, -141, -166, -193, -227, -262, -308, -298, -296, -206, -164, -133, -107, -96, -66, -48, 10, 51, 71, 157, 200, 260, 305, 408, 456, 510, 615, 698, 650, 499, 344, 285, 198, 119, -27, 79, -78, -187, -405

Offset: 1

Rémy Sigrist, Oct 27 2024

sign

			a(3) = A377388(1) + A377388(2) + A377388(3) = 0 + 1 - 2 = -1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 10000 terms
Rémy Sigrist, C++ program

Cf. A377388.

# uses A377388gen() in A377388
from itertools import accumulate, islice
def agen(): yield from accumulate(A377388gen())
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 27 2024

a(n) = Sum_{k = 1..n} A377388(n).

A377351 Lexicographically earliest sequence of positive integers such that the means of consecutive terms are all distinct.

Original entry on oeis.org

1, 2, 4, 7, 5, 10, 12, 18, 13, 16, 28, 23, 14, 25, 48, 32, 17, 46, 30, 45, 67, 60, 27, 71, 39, 85, 68, 99, 78, 44, 102, 87, 126, 57, 118, 69, 121, 74, 125, 119, 112, 136, 107, 110, 170, 120, 175, 142, 194, 75, 222, 152, 164, 180, 177, 184, 188, 135, 255, 210

Offset: 1

Rémy Sigrist, Oct 26 2024

nonn

In other words, for any distinct nonempty intervals t..u and v..w, Sum_{i = t..u} a(i)/(u-t+1) <> Sum_{j = v..w} a(j)/(w-v+1).
This sequence corresponds essentially to the first differences of A033808.
By necessity, all terms are distinct.

			The first terms, alongside the means of consecutive terms ending with a(n), are:
  n  a(n)  Corresponding means
  -  ----  ------------------------------------------
  1     1  1
  2     2  3/2, 2
  3     4  7/3, 3, 4
  4     7  7/2, 13/3, 11/2, 7
  5     5  19/5, 9/2, 16/3, 6, 5
  6    10  29/6, 28/5, 13/2, 22/3, 15/2, 10
  7    12  41/7, 20/3, 38/5, 17/2, 9, 11, 12
  8    18  59/8, 58/7, 28/3, 52/5, 45/4, 40/3, 15, 18

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program

Cf. A033808, A377388.

from fractions import Fraction
from itertools import count, islice
def agen(): # generator of terms
    alst, means_seen = [1], {1}
    while True:
        yield alst[-1]
        for k in count(1):
            if k in means_seen: continue
            mk, failed, sk = {k}, False, k
            for j in range(1, len(alst)+1):
                sk += alst[-j]
                m = Fraction(sk, j+1)
                if m in means_seen or m in mk: failed = True; break
                mk.add(m)
            if not failed: break
        means_seen |= mk
        alst.append(k)
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 26 2024, Oct 28 2024

a(n) = A033808(n) - A033808(n-1).

cp's OEIS Frontend

A377389 Partial sums of A377388.

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