A376590 - OEIS

A376590

Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

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, -1, 0, 2, -2, 3, -2, 0, 0, -1, 0, 1, -1, 2, -2, 0, 1, -1, 0, 1, -1, 2, -2, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 1, 2, -3, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 2, -2, 3, -2, -1

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OFFSET

1,5

LINKS

Table of n, a(n) for n=1..78.

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

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

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.

Zeros are A376591, complement A376592.

Sorted positions of first appearances are A376655.

A000040 lists the prime numbers, differences A001223.

A001597 lists perfect-powers, complement A007916.

A005117 lists squarefree numbers, complement A013929 (differences A078147).

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

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).

Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Sequence in context: A368753 A345907 A293019 * A087479 A263343 A326413

Adjacent sequences: A376587 A376588 A376589 * A376591 A376592 A376593

KEYWORD

sign

AUTHOR

Gus Wiseman, Oct 01 2024

STATUS

approved