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

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304

Offset: 1

Author

Gus Wiseman, Sep 30 2024

Keywords

• nonn

Comments

These are points at which the second differences are negative.

Perfect-powers (A001597) are numbers with a proper integer root.

Note that, for some sources, downward concavity is positive curvature.

From Robert Israel, Oct 31 2024: (Start)

The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.

The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.

The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Gus Wiseman, Points of downward concavity in the perfect-powers.

Crossrefs

The version for A000002 is A025505, complement A022297. See also A054354, A376604.

For first differences we have A053289, union A023055, firsts A376268, A376519.

For primes instead of perfect-powers we have A258026.

For upward concavity we have A376560 (probably the complement).

A000961 lists the prime-powers inclusive, exclusive A246655.

A001597 lists the perfect-powers.

A007916 lists the non-perfect-powers.

A112344 counts partitions into perfect-powers, factorizations A294068.

A333254 gives run-lengths of differences between consecutive primes.

Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

Programs

Maple

N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]