A375714 Positions of non-successions of consecutive non-perfect-powers. Numbers k such that the k-th non-perfect-power is at least two fewer than the next.

2, 5, 11, 19, 20, 24, 27, 39, 53, 69, 87, 107, 110, 112, 127, 151, 177, 196, 204, 221, 233, 265, 299, 317, 334, 372, 412, 454, 481, 497, 543, 591, 641, 693, 747, 803, 861, 921, 959, 982, 1046, 1112, 1180, 1250, 1284, 1321, 1395, 1471, 1549, 1629, 1675, 1710

Offset: 1

Author

Gus Wiseman, Sep 10 2024

Keywords

• nonn

Comments

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

Examples

			The initial non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, which increase by more than one after term 2, term 5, term 11, etc.
		

Crossrefs

First differences are A375702.

Positions of terms > 1 in A375706 (differences of A007916).

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

The complement is A375740.

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

Prime-powers inclusive:

- terms: A000961

- differences: A057820

- runs: A373675, A373673, A373674, A174965

- antiruns: A373576, A120430, A006549, A373671

Non-prime-powers inclusive:

- terms: A361102

- differences: A375708

- runs: A373678, A373676, A373677, A110969

- antiruns: A373679, A373575, A255346, A373672

Cf. A000040, A001223, A046933, A053289, A073783, A246655, A251092, A375734.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
ce=Select[Range[100],radQ];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

Python

from itertools import count, islice
from sympy import perfect_power
def A375714_gen(): # generator of terms
    a, b = -1, 0
    for n in count(1):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&(c^1):
            yield a
        b = c
A375714_list = list(islice(A375714_gen(),52)) # Chai Wah Wu, Sep 11 2024

Formula

A007916(a(n)+1) - A007916(a(n)) > 1.