cp's OEIS Frontend

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

%I A357579 #22 Oct 23 2022 04:33:00
%S A357579 2,3,7,5,6,12,10,11,17,18,15,13,20,14,23,19,28,26,22,21,29,33,35,37,
%T A357579 24,31,30,38,34,41,39,40,44,43,46,42,51,45,54,53,48,57,47,50,59,52,61,
%U A357579 58,55,60,56,66,67,65,62,70,63,69,73,72,76,74,68,79
%N A357579 Lexicographically earliest sequence of distinct numbers such that no sum of consecutive terms is a square or higher power of an integer.
%C A357579 This is inspired by sequence A254337, where sums equal to prime numbers are disallowed.
%C A357579 An unproved conjecture (for the present sequence) is that all integers which are not nontrivial powers will eventually appear.
%H A357579 Rémy Sigrist, <a href="/A357579/b357579.txt">Table of n, a(n) for n = 1..10000</a>
%H A357579 Rémy Sigrist, <a href="/A357579/a357579.gp.txt">PARI program</a>
%H A357579 Carl Witthoft, <a href="https://github.com/cellocgw/Rgoodies/blob/master/hasAnyPwr.R">R program</a>
%e A357579 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.
%o A357579 (R) # hasAnyPwr  and helper function are in the GitHub link
%o A357579 (Python)
%o A357579 def is_pow(n, k):
%o A357579     while n%k == 0: n = n//k
%o A357579     return n == 1
%o A357579 def any_power(n):
%o A357579     return any((is_pow(n, k) for k in range(2,1+n//2)))
%o A357579 terms,s,sums = [2,], set((2,)), set((2,))
%o A357579 for i in range(100):
%o A357579     t = 3
%o A357579     while t in s or any_power(t) or any((any_power(j + t) for j in sums)):
%o A357579         t+=1
%o A357579     s.add(t);terms.append(t)
%o A357579     sums = set(map(lambda k:k+t, sums))
%o A357579     sums.add(t)
%o A357579 print(terms) # _Gleb Ivanov_, Oct 07 2022
%o A357579 (PARI) See Links section.
%Y A357579 Cf. A254337.
%K A357579 nonn
%O A357579 1,1
%A A357579 _Carl Witthoft_, Oct 03 2022