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A373575 Numbers k such that k and k-1 both have at least two distinct prime factors. First element of the n-th maximal antirun of non-prime-powers.

%I A373575 #16 Feb 23 2025 12:53:45
%S A373575 1,15,21,22,34,35,36,39,40,45,46,51,52,55,56,57,58,63,66,69,70,75,76,
%T A373575 77,78,85,86,87,88,91,92,93,94,95,96,99,100,105,106,111,112,115,116,
%U A373575 117,118,119,120,123,124,130,133,134,135,136,141,142,143,144,145
%N A373575 Numbers k such that k and k-1 both have at least two distinct prime factors. First element of the n-th maximal antirun of non-prime-powers.
%C A373575 The last element of the same antirun is given by A255346.
%C A373575 An antirun of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by more than one.
%H A373575 Harvey P. Dale, <a href="/A373575/b373575.txt">Table of n, a(n) for n = 1..1000</a>
%e A373575 The maximal antiruns of non-prime-powers begin:
%e A373575    1   6  10  12  14
%e A373575   15  18  20
%e A373575   21
%e A373575   22  24  26  28  30  33
%e A373575   34
%e A373575   35
%e A373575   36  38
%e A373575   39
%e A373575   40  42  44
%e A373575   45
%e A373575   46  48  50
%t A373575 Select[Range[100],!PrimePowerQ[#]&&!PrimePowerQ[#-1]&]
%t A373575 Join[{1},SequencePosition[Table[If[PrimeNu[n]>1,1,0],{n,150}],{1,1}][[;;,2]]] (* _Harvey P. Dale_, Feb 23 2025 *)
%Y A373575 Runs of prime-powers:
%Y A373575 - length A174965
%Y A373575 - min A373673
%Y A373575 - max A373674
%Y A373575 - sum A373675
%Y A373575 Runs of non-prime-powers:
%Y A373575 - length A110969
%Y A373575 - min A373676
%Y A373575 - max A373677
%Y A373575 - sum A373678
%Y A373575 Antiruns of prime-powers:
%Y A373575 - length A373671
%Y A373575 - min A120430
%Y A373575 - max A006549
%Y A373575 - sum A373576
%Y A373575 Antiruns of non-prime-powers:
%Y A373575 - length A373672
%Y A373575 - min A373575 (this sequence)
%Y A373575 - max A255346
%Y A373575 - sum A373679
%Y A373575 A000961 lists all powers of primes. A246655 lists just prime-powers.
%Y A373575 A057820 gives first differences of consecutive prime-powers, gaps A093555.
%Y A373575 A356068 counts non-prime-powers up to n.
%Y A373575 A361102 lists all non-prime-powers (A024619 if not including 1).
%Y A373575 Various run-lengths: A053797, A120992, A175632, A176246.
%Y A373575 Various antirun-lengths: A027833, A373127, A373403, A373409.
%Y A373575 Cf. A001359, A008864, A014963, A067774, A251092, A373669, A373670.
%K A373575 nonn
%O A373575 1,2
%A A373575 _Gus Wiseman_, Jun 18 2024