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A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

%I A373669 #20 Jun 18 2024 08:19:11
%S A373669 1,5,7,12,18,190,28,109,40,28195574,53
%N A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.
%C A373669 A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
%C A373669 Are there only 9 terms?
%C A373669 From _David A. Corneth_, Jun 14 2024: (Start)
%C A373669 No. a(10) exists.
%C A373669 Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)
%H A373669 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373669 The maximal runs of non-prime-powers begin:
%e A373669    1
%e A373669    6
%e A373669   10
%e A373669   12
%e A373669   14  15
%e A373669   18
%e A373669   20  21  22
%e A373669   24
%e A373669   26
%e A373669   28
%e A373669   30
%e A373669   33  34  35  36
%e A373669   38  39  40
%e A373669   42
%e A373669   44  45  46
%e A373669   48
%e A373669   50  51  52
%e A373669   54  55  56  57  58
%e A373669   60
%t A373669 q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
%t A373669 spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
%t A373669 Table[Position[q,k][[1,1]],{k,spna[q]}]
%Y A373669 For composite runs we have A073051, sorted A373400, firsts of A176246.
%Y A373669 For squarefree runs we have firsts of A120992.
%Y A373669 For prime-powers runs we have firsts of A174965.
%Y A373669 For prime runs we have firsts of A251092 or A175632.
%Y A373669 For squarefree antiruns we have A373128, firsts of A373127.
%Y A373669 For nonsquarefree runs we have A373199, firsts of A053797.
%Y A373669 The sorted version is A373670.
%Y A373669 For antiruns we have firsts of A373672.
%Y A373669 For runs of non-prime-powers:
%Y A373669 - length A110969
%Y A373669 - min A373676
%Y A373669 - max A373677
%Y A373669 - sum A373678
%Y A373669 A000961 lists the powers of primes (including 1).
%Y A373669 A005117 lists the squarefree numbers, first differences A076259.
%Y A373669 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A373669 A057820 gives first differences of consecutive prime-powers, gaps A093555.
%Y A373669 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A373669 A361102 lists the non-prime-powers, without 1 A024619.
%Y A373669 Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.
%K A373669 nonn,more
%O A373669 1,2
%A A373669 _Gus Wiseman_, Jun 14 2024