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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.
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%I #9 Sep 13 2024 06:58:11

%S 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,

%T 33,34,35,36,37,38,40,41,42,43,44,45,46,47,48,49,50,51,52,54,55,56,57,

%U 58,59,60,61,62,63,64,65,66,67,68,70,71,72,73,74,75,76,77

%N 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.

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

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

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

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

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Join@@Position[Differences[Select[Range[100],radQ]],1]

%o (Python)

%o from itertools import count, islice

%o from sympy import perfect_power

%o def A375740_gen(): # generator of terms

%o a, b = -1, 0

%o for n in count(2):

%o c = not perfect_power(n)

%o if c:

%o a += 1

%o if b&c:

%o yield a

%o b = c

%o A375740_list = list(islice(A375740_gen(), 52)) # _Chai Wah Wu_, Sep 11 2024

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

%Y The complement is A375714, differences A375702.

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

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

%Y A000040 lists the prime numbers, differences A001223.

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

%Y A001597 lists perfect-powers, differences A053289.

%Y A002808 lists the composite numbers, differences A073783.

%Y A018252 lists the nonprime numbers, differences A065310.

%Y Non-perfect-powers:

%Y - terms: A007916

%Y - differences: A375706

%Y - runs: sum A375705, A375703, A375704, A375702

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

%Y Non-prime-powers (exclusive):

%Y - terms: A361102

%Y - differences: A375708

%Y - runs: A373678, A373676, A373677, A110969

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 10 2024