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A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

%I A376561 #15 Nov 02 2024 09:11:13
%S A376561 2,5,7,13,14,18,19,21,24,25,29,30,39,40,45,51,52,56,59,66,70,71,74,87,
%T A376561 94,101,102,108,110,112,113,119,127,135,143,144,156,157,160,161,169,
%U A376561 178,187,196,205,206,215,224,225,234,244,263,273,283,284,293,294,304
%N A376561 Points of downward concavity in the sequence of perfect-powers (A001597).
%C A376561 These are points at which the second differences are negative.
%C A376561 Perfect-powers (A001597) are numbers with a proper integer root.
%C A376561 Note that, for some sources, downward concavity is positive curvature.
%C A376561 From _Robert Israel_, Oct 31 2024: (Start)
%C A376561 The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
%C A376561 The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
%C A376561 The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)
%H A376561 Robert Israel, <a href="/A376561/b376561.txt">Table of n, a(n) for n = 1..10000</a>
%H A376561 Gus Wiseman, <a href="/A376561/a376561.png">Points of downward concavity in the perfect-powers</a>.
%e A376561 The perfect powers (A001597) are:
%e A376561   1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
%e A376561 with first differences (A053289):
%e A376561   3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
%e A376561 with first differences (A376559):
%e A376561   1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
%e A376561 with negative positions (A376561):
%e A376561   2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...
%p A376561 N:= 10^6: # to use perfect powers <= N
%p A376561 P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
%p A376561 P:= sort(convert(P,list)):
%p A376561 select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # _Robert Israel_, Oct 31 2024
%t A376561 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A376561 Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]
%Y A376561 The version for A000002 is A025505, complement A022297. See also A054354, A376604.
%Y A376561 For first differences we have A053289, union A023055, firsts A376268, A376519.
%Y A376561 For primes instead of perfect-powers we have A258026.
%Y A376561 For upward concavity we have A376560 (probably the complement).
%Y A376561 A000961 lists the prime-powers inclusive, exclusive A246655.
%Y A376561 A001597 lists the perfect-powers.
%Y A376561 A007916 lists the non-perfect-powers.
%Y A376561 A112344 counts partitions into perfect-powers, factorizations A294068.
%Y A376561 A333254 gives run-lengths of differences between consecutive primes.
%Y A376561 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 A376561 Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.
%K A376561 nonn
%O A376561 1,1
%A A376561 _Gus Wiseman_, Sep 30 2024