A251092 - OEIS

%I #41 Jun 29 2021 13:34:21

%S 3,2,2,1,2,1,2,1,1,2,1,2,1,1,1,1,2,2,1,1,1,2,2,1,1,1,1,2,2,2,1,1,2,1,

%T 2,1,1,1,2,1,2,1,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,2,1,1,1,

%U 1,1,1,1,2,1,1,1,1,2,1,1,1,2,1,1,2,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1

%N a(n) is the number of primes in the n-th group of consecutive primes among the odd numbers.

%C Explanation:

%C 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,... = Odd numbers

%C    ^  ^  ^      ^   ^       ^   ^       ^         = Prime numbers

%C   <---3--->    <--2-->     <--2-->    <-1->       = Primes beside other primes divided into groups of how many there are.

%C Note that the first group (3, 5 and 7) is the only group of 3. Therefore the rest of the sequence only consists of 1's and 2's.

%C Essentially the same as A175632. - _Robert Israel_, Mar 29 2015

%H Harvey P. Dale, <a href="/A251092/b251092.txt">Table of n, a(n) for n = 1..1000</a>

%p N:= 1000: # to use the first N+1 odd numbers

%p L:= map(t -> isprime(2*t+1), [$1..N]):

%p Starts:= [1, op(select(i -> L[i] and not L[i-1], [$2..N]))]:

%p Ends:= select(i -> L[i] and not L[i+1], [$1..N-1]):

%p seq(Ends[i]-Starts[i]+1,i=1..nops(Ends)); # _Robert Israel_, Mar 27 2015

%t Length /@ Split[Select[2 Range@ 1200 - 1, PrimeQ], #2 - #1 == 2 &] (* _Michael De Vlieger_, Mar 20 2015 *)

%t Length/@DeleteCases[Split[Table[If[PrimeQ[n],1,0],{n,3,1001,2}]],_?(FreeQ[ #,1]&)] (* _Harvey P. Dale_, Jun 29 2021 *)

%Y Cf. A000040 (prime numbers), A005408 (odd numbers), A001097 (twin primes), A175632.

%K nonn

%O 1,1

%A _Arthur Edward Chadwick_, Mar 20 2015