%I #10 Oct 02 2024 11:36:48

%S 1,-1,0,2,-2,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,1,-1,0,1,-1,0,0,0,0,

%T 0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

%C Non-perfect-powers (A007916) are numbers without a proper integer root.

%e The non-perfect powers (A007916) are:

%e 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...

%e with first differences (A375706):

%e 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...

%e with first differences (A376562):

%e 1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Differences[Select[Range[100],radQ],2]

%o (Python)

%o from itertools import count

%o from sympy import mobius, integer_nthroot, perfect_power

%o def A376562(n):

%o def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))

%o m, k = n, f(n)

%o while m != k: m, k = k, f(k)

%o r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)

%o return next(i for i in count(1-k) if not perfect_power(r+i)) # _Chai Wah Wu_, Oct 02 2024

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

%Y For first differences we had A375706, ones A375740, complement A375714.

%Y Positions of zeros are A376588, complement A376589.

%Y Runs of non-perfect-powers:

%Y - length: A375702 = A053289(n+1) - 1

%Y - first: A375703 (same as A216765 with 2 exceptions)

%Y - last: A375704 (same as A045542 with 8 removed)

%Y - sum: A375705

%Y A000961 lists prime-powers inclusive, exclusive A246655.

%Y A007916 lists non-perfect-powers, complement A001597.

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

%Y A333254 gives run-lengths of differences between consecutive primes.

%Y For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).

%Y For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

%Y Cf. A025475, A052410, A053707, A064113, A069623, A093555, A174965, A182853, A336416, A336417, A361102.

%K sign

%O 1,4

%A _Gus Wiseman_, Oct 01 2024