cp's OEIS Frontend

A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

%I A373670 #6 Jun 15 2024 22:50:29
%S A373670 1,5,7,12,18,28,40,53,71,109,170,190,198,207,236,303,394,457,606,774,
%T A373670 1069,1100,1225,1881,1930,1952,2247,2281,3140,3368,3451,3493,3713,
%U A373670 3862,4595,4685,6625,8063,8121,8783,12359,12650,14471,14979,15901,17129,19155
%N A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.
%C A373670 The unsorted version is A373669.
%e A373670 The maximal runs of non-prime-powers begin:
%e A373670    1
%e A373670    6
%e A373670   10
%e A373670   12
%e A373670   14  15
%e A373670   18
%e A373670   20  21  22
%e A373670   24
%e A373670   26
%e A373670   28
%e A373670   30
%e A373670   33  34  35  36
%e A373670   38  39  40
%e A373670   42
%e A373670   44  45  46
%e A373670   48
%e A373670   50  51  52
%e A373670   54  55  56  57  58
%e A373670   60
%e A373670 So the a(n)-th runs begin:
%e A373670    1
%e A373670   14  15
%e A373670   20  21  22
%e A373670   33  34  35  36
%e A373670   54  55  56  57  58
%t A373670 t=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&];
%t A373670 Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
%Y A373670 For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.
%Y A373670 For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.
%Y A373670 For composite runs we have A373400, unsorted A073051, firsts of A176246.
%Y A373670 For prime antiruns we have A373402.
%Y A373670 For runs of non-prime-powers:
%Y A373670 - length A110969, firsts A373669, sorted A373670 (this sequence):
%Y A373670 - min A373676
%Y A373670 - max A373677
%Y A373670 - sum A373678
%Y A373670 For runs of prime-powers:
%Y A373670 - length A174965
%Y A373670 - min A373673
%Y A373670 - max A373674
%Y A373670 - sum A373675
%Y A373670 A000961 lists the powers of primes (including 1).
%Y A373670 A057820 gives first differences of consecutive prime-powers, gaps A093555.
%Y A373670 A361102 lists the non-prime-powers, without 1 A024619.
%Y A373670 Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.
%K A373670 nonn
%O A373670 1,2
%A A373670 _Gus Wiseman_, Jun 15 2024