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approved

editing

proposed

def iterfun(f, n=0):

return m

return m def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

return a+iterfun(lambda x:f(x)+2, b)-(b<<1) # Chai Wah Wu, Oct 02 2024

(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

approved

editing

proposed

approved

editing

proposed

Zeros are A376591, non-zeros complement A376592.

`A001597 lists perfect-powers, complement A007916.

A005117 lists squarefree numbers, complement A013929 (differences A076259A078147).

A013929 lists nonsquarefree numbers, differences A078147.

`A053707 gives first differences of consecutive nonprime prime-powers.

`A064113 lists positions of adjacent equal prime gaps.

A112344 A073576 counts integer partitions into perfect-powers, squarefree numbers, factorizations A294068A050320.

`A246655 lists prime-powers exclusive, inclusive A000961.

Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, `A174965, A182853, A251092, A373197, A373198, A375707, A376342.

A007916 `A001597 lists non-perfect-powers, complement A007916.

A005117 lists squarefree numbers, differences A076259.

`A053707 gives first differences of consecutive nonprime prime -powers.

`A064113 lists positions of adjacent equal prime gaps.

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

`A246655 lists prime-powers exclusive, inclusive A000961.

A336417 counts perfect-power divisors of A006939.

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

For squarefree numbers: A005117 (terms), A076259 (first differences), A376590 (second differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).

Cf. `A025475, `A069623, `A258025, `A258026.

Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, `A174965, A182853, A251092, `A375702, `A376308A373197, A373198, A375707, A376342.

Cf. A007674, A053797, `A053806, A061398, A072284, A112925, A112926, A120992, A373197, A373198, A375707, `A376312, `A376342.

Cf. A007674 max_antrun_sqf, , A053797 nonsqf_runlens, , `A053806 min_nonsqf_run, , A061398 num_sqf_btwn_pri, , A072284 min_run_sqf, , A112925 max_sqf_less_pri, , A112926 min_sqf_grtr_pri, , A120992 sqf_runlens, , A373197 sqfsum_btwn_pri, , A373198 sqflen_btwn_pri, , A375707 sqf_btwn_nonsqf, , `A376312 nonsqf_diffs_compress, , `A376342 sqf_diffs_compress_pos1.

allocated for Gus Wiseman

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

1,5

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, ...

Differences[Select[Range[100], SquareFreeQ], 2]

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, non-zeros A376592.

Sorted positions of first appearances are A376655.

A000040 lists the prime numbers, differences A001223.

A007916 lists non-perfect-powers.

A013929 lists nonsquarefree numbers, differences A078147.

A053707 gives first differences of consecutive nonprime prime powers.

A064113 lists positions of adjacent equal prime gaps.

A112344 counts partitions into perfect-powers, factorizations A294068.

A246655 lists prime-powers exclusive, inclusive A000961.

A333254 lists run-lengths of differences between consecutive primes.

A336417 counts perfect-power divisors of A006939.

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

For squarefree numbers: A005117 (terms), A076259 (first differences), A376590 (second differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).

Cf. `A025475, `A069623, `A258025, `A258026.

Cf. `A174965, A182853, A251092, `A375702, `A376308.

Cf. A007674 max_antrun_sqf, A053797 nonsqf_runlens, A053806 min_nonsqf_run, A061398 num_sqf_btwn_pri, A072284 min_run_sqf, A112925 max_sqf_less_pri, A112926 min_sqf_grtr_pri, A120992 sqf_runlens, A373197 sqfsum_btwn_pri, A373198 sqflen_btwn_pri, A375707 sqf_btwn_nonsqf, A376312 nonsqf_diffs_compress, A376342 sqf_diffs_compress_pos1.

allocated

sign

Gus Wiseman, Oct 01 2024

approved

editing