A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, -16, -1, 21, 2, 2, -15, -11, 30, 2, 2, 2, 2, 2, 2, 2, -22, -15, 41, 2, 2, 2, -36, 3, 37, 2, 2, 2, -34, -11, 49, 2, 2, -66, 45, 3, -61, 2, 83, 2, 2, 2, 2, -63, 25, 42, 2, -9, -89

Offset: 1

Author

Gus Wiseman, Sep 28 2024

Keywords

• sign

Comments

Perfect-powers A007916 are numbers with a proper integer root.

Does this sequence contain zero?

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, ...
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, ...
		

Crossrefs

The version for A000002 is A376604, first differences of A054354.

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

A000961 lists prime-powers inclusive, exclusive A246655.

A001597 lists perfect-powers, complement A007916.

A112344 counts integer partitions into perfect-powers, factorizations A294068.

For perfect-powers: A053289 (first differences), A376560 (positive curvature), A376561 (negative curvature).

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

Cf. A045542, A052410, A053707, A064113, A069623, A174965, A216765, A251092, A333254, A336416, A361102.

Programs

Mathematica

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

PARI

lista(nn) = my(v = concat (1, select(ispower, [1..nn])), w = vector(#v-1, i, v[i+1] - v[i])); vector(#w-1, i, w[i+1] - w[i]); \\ Michel Marcus, Oct 02 2024

Python

from sympy import mobius, integer_nthroot
def A376559(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = bisection(f,n,n)
    b = bisection(lambda x:f(x)+1,a,a)
    return a+bisection(lambda x:f(x)+2,b,b)-(b<<1) # Chai Wah Wu, Oct 02 2024