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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.

%I A375740 #9 Sep 13 2024 06:58:11
%S A375740 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 A375740 33,34,35,36,37,38,40,41,42,43,44,45,46,47,48,49,50,51,52,54,55,56,57,
%U A375740 58,59,60,61,62,63,64,65,66,67,68,70,71,72,73,74,75,76,77
%N 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.
%C A375740 Positions in A007916 of numbers k such that k+1 is also a member.
%C A375740 Positions of 1's in A375706 (first differences of A007916).
%C A375740 Non-perfect-powers (A007916) are numbers with no proper integer roots.
%e A375740 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 A375740 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375740 Join@@Position[Differences[Select[Range[100],radQ]],1]
%o A375740 (Python)
%o A375740 from itertools import count, islice
%o A375740 from sympy import perfect_power
%o A375740 def A375740_gen(): # generator of terms
%o A375740     a, b = -1, 0
%o A375740     for n in count(2):
%o A375740         c = not perfect_power(n)
%o A375740         if c:
%o A375740             a += 1
%o A375740         if b&c:
%o A375740             yield a
%o A375740     b = c
%o A375740 A375740_list = list(islice(A375740_gen(), 52)) # _Chai Wah Wu_, Sep 11 2024
%Y A375740 The version for non-prime-powers is A375713, differences A373672.
%Y A375740 The complement is A375714, differences A375702.
%Y A375740 The version for prime-powers is A375734, differences A373671.
%Y A375740 The complement for non-prime-powers is A375928, differences A110969.
%Y A375740 A000040 lists the prime numbers, differences A001223.
%Y A375740 A000961 lists prime-powers (inclusive), differences A057820.
%Y A375740 A001597 lists perfect-powers, differences A053289.
%Y A375740 A002808 lists the composite numbers, differences A073783.
%Y A375740 A018252 lists the nonprime numbers, differences A065310.
%Y A375740 Non-perfect-powers:
%Y A375740 - terms: A007916
%Y A375740 - differences: A375706
%Y A375740 - runs: sum A375705, A375703, A375704, A375702
%Y A375740 - anti-runs: A375737, A375738, A375739, A375736.
%Y A375740 Non-prime-powers (exclusive):
%Y A375740 - terms: A361102
%Y A375740 - differences: A375708
%Y A375740 - runs: A373678, A373676, A373677, A110969
%Y A375740 - anti-runs: A373679, A373575, A255346, A373672
%Y A375740 Cf. A006549, A046933, A093555, A174965, A246655, A251092.
%K A375740 nonn
%O A375740 1,2
%A A375740 _Gus Wiseman_, Sep 10 2024