A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

-2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0

Offset: 1

Author

Gus Wiseman, Oct 02 2024

Keywords

• sign

Comments

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

Examples

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
		

Crossrefs

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

For first differences we had A375735, ones A375713(n) - 1.

Positions of zeros are A376600, complement A376601.

A000961 lists prime-powers inclusive, exclusive A246655.

A007916 lists non-perfect-powers.

A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.

A321346/A321378 count integer partitions without prime-powers, factorizations A322452.

For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).

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

Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

Mathematica

Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]

Python

from sympy import primepi, integer_nthroot
def A376599(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024