A357579 Lexicographically earliest sequence of distinct numbers such that no sum of consecutive terms is a square or higher power of an integer.

2, 3, 7, 5, 6, 12, 10, 11, 17, 18, 15, 13, 20, 14, 23, 19, 28, 26, 22, 21, 29, 33, 35, 37, 24, 31, 30, 38, 34, 41, 39, 40, 44, 43, 46, 42, 51, 45, 54, 53, 48, 57, 47, 50, 59, 52, 61, 58, 55, 60, 56, 66, 67, 65, 62, 70, 63, 69, 73, 72, 76, 74, 68, 79

Offset

1,1

Comments

This is inspired by sequence A254337, where sums equal to prime numbers are disallowed.

An unproved conjecture (for the present sequence) is that all integers which are not nontrivial powers will eventually appear.

Links

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

Rémy Sigrist, PARI program

Carl Witthoft, R program

Example

Clearly 0 and 1 are powers of themselves so they are rejected. 2 is the first term. Then neither 3 nor (3+2) is a power so 3 is accepted. 4 is a power and thus rejected. (5+3) is 2^3, so reject (for now) 5. Same for 6; (7+3) and (7+3+2) are not powers, so 7 is accepted.

Prog

(R) # hasAnyPwr  and helper function are in the GitHub link
(Python)
def is_pow(n, k):
    while n%k == 0: n = n//k
    return n == 1
def any_power(n):
    return any((is_pow(n, k) for k in range(2, 1+n//2)))
terms, s, sums = [2, ], set((2, )), set((2, ))
for i in range(100):
    t = 3
    while t in s or any_power(t) or any((any_power(j + t) for j in sums)):
        t+=1
    s.add(t); terms.append(t)
    sums = set(map(lambda k:k+t, sums))
    sums.add(t)
print(terms) # Gleb Ivanov, Oct 07 2022
(PARI) See Links section.

Crossrefs

Cf. A254337.

Sequence in context: A235801 A076986 A333386 * A334126 A341717 A225403

Adjacent sequences: A357576 A357577 A357578 * A357580 A357581 A357582

Keyword

nonn

Author

Carl Witthoft, Oct 03 2022

Status

approved