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User: Arthur Edward Chadwick

Arthur Edward Chadwick has authored 1 sequences.

A251092 a(n) is the number of primes in the n-th group of consecutive primes among the odd numbers.

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, 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, 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

Offset: 1

Arthur Edward Chadwick, Mar 20 2015

nonn

Explanation:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,... = Odd numbers
   ^  ^  ^      ^   ^       ^   ^       ^         = Prime numbers
  <---3--->    <--2-->     <--2-->    <-1->       = Primes beside other primes divided into groups of how many there are.
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.
Essentially the same as A175632. - Robert Israel, Mar 29 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

N:= 1000: # to use the first N+1 odd numbers
L:= map(t -> isprime(2*t+1), [$1..N]):
Starts:= [1, op(select(i -> L[i] and not L[i-1], [$2..N]))]:
Ends:= select(i -> L[i] and not L[i+1], [$1..N-1]):
seq(Ends[i]-Starts[i]+1,i=1..nops(Ends)); # Robert Israel, Mar 27 2015

Length /@ Split[Select[2 Range@ 1200 - 1, PrimeQ], #2 - #1 == 2 &] (* Michael De Vlieger, Mar 20 2015 *)
Length/@DeleteCases[Split[Table[If[PrimeQ[n],1,0],{n,3,1001,2}]],?(FreeQ[ #,1]&)] (* _Harvey P. Dale, Jun 29 2021 *)

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User: Arthur Edward Chadwick

A251092 a(n) is the number of primes in the n-th group of consecutive primes among the odd numbers.

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