A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, 1, 0, 1, -2, 4, -4, 0, 4, 2, -4, -2, 2, -2, 2, 4, -4, -2, -1, 2, 3, -4, 8, -8, 4, 0, -2, -2, 2, 2, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 0, 6, -3, -4, 5, 0, -4, 4, -2, -2

Offset: 1

Author

Gus Wiseman, Oct 02 2024

Keywords

• sign

Comments

For the exclusive version, shift left once.

Examples

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
		

Crossrefs

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

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).

Positions of zeros are A376597, complement A376598.

Sorted positions of first appearances are A376653, exclusive A376654.

A000961 lists prime-powers inclusive, exclusive A246655.

A001597 lists perfect-powers, complement A007916.

A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

A064113 lists positions of adjacent equal prime gaps.

For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).

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

Cf. A025475, A045542, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Programs

Mathematica

Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2]

Python

from sympy import primepi, integer_nthroot
def A376596(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+x-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