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A373678 Sums of maximal runs of non-prime-powers.

%I A373678 #10 Jun 17 2024 14:48:34
%S A373678 1,6,10,12,29,18,63,24,26,28,30,138,117,42,135,48,153,280,60,125,131,
%T A373678 207,72,380,80,82,430,651,297,102,315,108,333,819,369,126,259,670,138,
%U A373678 1296,150,770,800,495,168,513,880,180,1674,192,585,198,2255,2387,675
%N A373678 Sums of maximal runs of non-prime-powers.
%C A373678 We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.
%C A373678 A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
%H A373678 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373678 The maximal runs of non-powers of primes begin:
%e A373678    1
%e A373678    6
%e A373678   10
%e A373678   12
%e A373678   14  15
%e A373678   18
%e A373678   20  21  22
%e A373678   24
%e A373678   26
%e A373678   28
%e A373678   30
%e A373678   33  34  35  36
%e A373678   38  39  40
%e A373678   42
%e A373678   44  45  46
%e A373678   48
%e A373678   50  51  52
%e A373678   54  55  56  57  58
%e A373678   60
%t A373678 Total/@Split[Select[Range[100],!PrimePowerQ[#]&],#1+1==#2&]//Most
%Y A373678 A000040 lists the primes, differences A001223.
%Y A373678 A000961 lists all powers of primes (A246655 if not including 1).
%Y A373678 A025528 counts prime-powers up to n.
%Y A373678 A057820 gives first differences of consecutive prime-powers, gaps A093555.
%Y A373678 A361102 lists all non-prime-powers (A024619 if not including 1).
%Y A373678 See link for composite, prime, nonsquarefree, and squarefree runs.
%Y A373678 Prime-power runs: A373675, min A373673, max A373674, length A174965.
%Y A373678 Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
%Y A373678 Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
%Y A373678 Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
%Y A373678 Cf. A005117, A014963, A038664, A046933, A054265, A065855, A071148, A293697, A371201, A373400.
%K A373678 nonn
%O A373678 1,2
%A A373678 _Gus Wiseman_, Jun 16 2024