A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Gus Wiseman, Sep 10 2024

Keywords

• nonn

Comments

Positions in A007916 of numbers k such that k+1 is also a member.

Positions of 1's in A375706 (first differences of A007916).

Non-perfect-powers (A007916) are numbers with no proper integer roots.

Examples

			The non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, ... which increase by one after positions 1, 3, 4, 6, ...
		

Crossrefs

The version for non-prime-powers is A375713, differences A373672.

The complement is A375714, differences A375702.

The version for prime-powers is A375734, differences A373671.

The complement for non-prime-powers is A375928, differences A110969.

A000040 lists the prime numbers, differences A001223.

A000961 lists prime-powers (inclusive), differences A057820.

A001597 lists perfect-powers, differences A053289.

A002808 lists the composite numbers, differences A073783.

A018252 lists the nonprime numbers, differences A065310.

Non-perfect-powers:

- terms: A007916

- differences: A375706

- runs: sum A375705, A375703, A375704, A375702

- anti-runs: A375737, A375738, A375739, A375736.

Non-prime-powers (exclusive):

- terms: A361102

- differences: A375708

- runs: A373678, A373676, A373677, A110969

- anti-runs: A373679, A373575, A255346, A373672

Cf. A006549, A046933, A093555, A174965, A246655, A251092.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]

Python

from itertools import count, islice
from sympy import perfect_power
def A375740_gen(): # generator of terms
    a, b = -1, 0
    for n in count(2):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&c:
            yield a
    b = c
A375740_list = list(islice(A375740_gen(), 52)) # Chai Wah Wu, Sep 11 2024