OFFSET
1,5
EXAMPLE
The squarefree numbers (A005117) are:
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
MATHEMATICA
Differences[Select[Range[100], SquareFreeQ], 2]
PROG
(Python)
from math import isqrt
from sympy import mobius
def A376590(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 n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
a = iterfun(f, n)
b = iterfun(lambda x:f(x)+1, a)
return a+iterfun(lambda x:f(x)+2, b)-(b<<1) # Chai Wah Wu, Oct 02 2024
CROSSREFS
Sorted positions of first appearances are A376655.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree numbers: A076259 (first differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).
KEYWORD
sign
AUTHOR
Gus Wiseman, Oct 01 2024
STATUS
approved