login
Length of the runs of ones in A014963.
41

%I #15 Jun 18 2024 20:37:01

%S 1,1,1,1,2,1,3,1,1,1,1,4,3,1,3,1,3,5,1,2,2,3,1,5,1,1,5,7,3,1,3,1,3,7,

%T 3,1,2,5,1,9,1,5,5,3,1,3,5,1,9,1,3,1,11,11,3,1,3,5,1,1,7,4,5,5,1,5,3,

%U 1,5,3,13,3,1,3,13,5,5,3,1,3,5,1,5,5,5,3,5,7,3,7

%N Length of the runs of ones in A014963.

%C Unbounded sequence.

%C From A373669 we see that 10 first appears at a(28195574) = 10.

%C Also run-lengths of non-prime-powers (assuming 1 is not a prime-power), where a run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one. Also nonzero differences of consecutive prime-powers minus one. - _Gus Wiseman_, Jun 18 2024

%H Andrew Howroyd, <a href="/A110969/b110969.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5)=2 because the fifth run of ones in A014963 is of length 2.

%t Length /@ SplitBy[Table[Exp[MangoldtLambda[n]], {n, 400}], # != 1 &][[ ;; -1 ;; 2]] (* _Michael De Vlieger_, Mar 21 2024 *)

%t DeleteCases[Differences[Select[Range[100],PrimePowerQ]]-1,0] (* _Gus Wiseman_, Jun 18 2024 *)

%o (PARI) \\ b(n) returns boolean of A014963(n) == 1.

%o b(n)={my(t); !isprime(if(ispower(n, ,&t), t, n))}

%o seq(n)={my(k=1, i=0, L=List()); while(#L<n, i++; if(!b(i), if(i>k, listput(L, i-k)); k = i+1)); Vec(L)} \\ _Andrew Howroyd_, Jan 02 2020

%Y Cf. A014963.

%Y Positions of first appearances are A373670, sorted A373669.

%Y For runs of prime-powers:

%Y - length A174965, antiruns A373671

%Y - min A373673, antiruns A120430

%Y - max A373674, antiruns A006549

%Y - sum A373675, antiruns A373576

%Y For runs of non-prime-powers:

%Y - length A110969 (this sequence), antiruns A373672

%Y - min A373676, antiruns A373575

%Y - max A373677, antiruns A255346

%Y - sum A373678, antiruns A373679

%Y A000961 lists all powers of primes. A246655 lists just prime-powers.

%Y A057820 gives first differences of consecutive prime-powers, gaps A093555.

%Y A356068 counts non-prime-powers up to n.

%Y A361102 lists all non-prime-powers (A024619 if not including 1).

%Y Various run-lengths: A053797, A120992, A175632, A176246.

%Y Various antirun-lengths: A027833, A373127, A373403, A373409.

%Y Cf. A001359, A008864, A067774, A251092.

%K nonn

%O 1,5

%A _Franz Vrabec_, Sep 27 2005

%E Terms a(41) and beyond from _Andrew Howroyd_, Jan 02 2020