cp's OEIS Frontend

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.

%I A375714 #10 Sep 11 2024 11:03:09
%S A375714 2,5,11,19,20,24,27,39,53,69,87,107,110,112,127,151,177,196,204,221,
%T A375714 233,265,299,317,334,372,412,454,481,497,543,591,641,693,747,803,861,
%U A375714 921,959,982,1046,1112,1180,1250,1284,1321,1395,1471,1549,1629,1675,1710
%N 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.
%C A375714 Non-perfect-powers (A007916) are numbers with no proper integer roots.
%F A375714 A007916(a(n)+1) - A007916(a(n)) > 1.
%e A375714 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.
%t A375714 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375714 ce=Select[Range[100],radQ];
%t A375714 Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]
%o A375714 (Python)
%o A375714 from itertools import count, islice
%o A375714 from sympy import perfect_power
%o A375714 def A375714_gen(): # generator of terms
%o A375714     a, b = -1, 0
%o A375714     for n in count(1):
%o A375714         c = not perfect_power(n)
%o A375714         if c:
%o A375714             a += 1
%o A375714         if b&(c^1):
%o A375714             yield a
%o A375714         b = c
%o A375714 A375714_list = list(islice(A375714_gen(),52)) # _Chai Wah Wu_, Sep 11 2024
%Y A375714 First differences are A375702.
%Y A375714 Positions of terms > 1 in A375706 (differences of A007916).
%Y A375714 The complement for non-prime-powers is A375713, differences A373672.
%Y A375714 The complement is A375740.
%Y A375714 The version for non-prime-powers is A375928, differences A110969.
%Y A375714 Prime-powers inclusive:
%Y A375714 - terms: A000961
%Y A375714 - differences: A057820
%Y A375714 - runs: A373675, A373673, A373674, A174965
%Y A375714 - antiruns: A373576, A120430, A006549, A373671
%Y A375714 Non-prime-powers inclusive:
%Y A375714 - terms: A361102
%Y A375714 - differences: A375708
%Y A375714 - runs: A373678, A373676, A373677, A110969
%Y A375714 - antiruns: A373679, A373575, A255346, A373672
%Y A375714 Cf. A000040, A001223, A046933, A053289, A073783, A246655, A251092, A375734.
%K A375714 nonn
%O A375714 1,1
%A A375714 _Gus Wiseman_, Sep 10 2024