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A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.

%I A376560 #19 Dec 01 2024 03:42:53
%S A376560 1,3,4,6,8,9,10,11,12,15,16,17,20,22,23,26,27,28,31,32,33,34,35,36,37,
%T A376560 38,41,42,43,44,46,47,48,49,50,53,54,55,57,58,60,61,62,63,64,65,67,68,
%U A376560 69,72,73,75,76,77,78,79,80,81,82,83,84,85,86,88,89,90,91
%N A376560 Points of upward concavity in the sequence of perfect-powers (A001597). Positives of A376559.
%C A376560 These are points at which the second differences are positive.
%C A376560 Perfect-powers (A001597) are numbers with a proper integer root.
%C A376560 Note that, for some sources, upward concavity is negative curvature.
%H A376560 Robert Israel, <a href="/A376560/b376560.txt">Table of n, a(n) for n = 1..10000</a>
%H A376560 Gus Wiseman, <a href="/A376560/a376560.png">Points of upward concavity in the perfect-powers</a>.
%e A376560 The perfect powers (A001597) are:
%e A376560   1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
%e A376560 with first differences (A053289):
%e A376560   3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
%e A376560 with first differences (A376559):
%e A376560   1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
%e A376560 with positive positions (A376560):
%e A376560   1, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, ...
%p A376560 N:= 10^6: # to use perfect powers <= N
%p A376560 S:= {1,seq(seq(i^j,j=2..floor(log[i](N))),i=2..isqrt(N))}:
%p A376560 L:= sort(convert(S,list)):
%p A376560 DL:= L[2..-1]-L[1..-2]:
%p A376560 D2L:= DL[2..-1]-DL[1..-2]:
%p A376560 select(i -> D2L[i]>0, [$1..nops(D2L)]); # _Robert Israel_, Dec 01 2024
%t A376560 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A376560 Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],1]
%Y A376560 The version for A000002 is A022297, complement A025505. See also A054354, A376604.
%Y A376560 For first differences we have A053289, union A023055, firsts A376268, A376519.
%Y A376560 For primes instead of perfect-powers we have A258025.
%Y A376560 These are positions of positive terms in A376559.
%Y A376560 For downward concavity we have A376561 (probably the complement).
%Y A376560 A001597 lists the perfect-powers.
%Y A376560 A064113 lists positions of adjacent equal prime gaps.
%Y A376560 A333254 gives run-lengths of differences between consecutive primes.
%Y A376560 Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
%Y A376560 Cf. A006549, A025475, A030173, A045542, A052410, A053707, A054819, A069623, A174965, A216765, A251092, A376308.
%K A376560 nonn
%O A376560 1,2
%A A376560 _Gus Wiseman_, Sep 30 2024