OFFSET
1,2
COMMENTS
Split the odd nonprime sequence A014076 into two subsequences A091113 and A091236 with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 1 mod 4 occurs before interruption of a nonprime congruent to 3 mod 4.
Otherwise said: count the nonprimes congruent to 1 mod 4 such that the next larger and next smaller odd nonprime is congruent to 3 mod 4. - M. F. Hasler, Sep 30 2018
EXAMPLE
a(3) = 74 because 74 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4.
Below 10^2 = 100, there are only a(2) = 3 isolated odd nonprimes congruent to 1 mod 4: 33, 57 and 93. (Credits: Peter Munn, SeqFan list.) - M. F. Hasler, Sep 30 2018
MAPLE
A014076 := proc(n)
option remember;
if n = 1 then
1;
else
for a from procname(n-1)+2 by 2 do
if not isprime(a) then
return a;
end if;
end do:
end if;
end proc:
isA091113 := proc(n)
option remember;
if modp(n, 4) = 1 and not isprime(n) then
true;
else
false;
end if;
end proc:
isA091236 := proc(n)
option remember;
if modp(n, 4) = 3 and not isprime(n) then
true;
else
false;
end if;
end proc:
ct := 0 :
n := 1 :
for i from 2 do
odnpr := A014076(i) ;
prev := A014076(i-1) ;
nxt := A014076(i+1) ;
if isA091113(odnpr) and isA091236(prev) and isA091236(nxt) then
ct := ct+1 ;
end if;
if odnpr< 10^n and nxt >= 10^n then
print(n, ct) ;
n := n+1 ;
end if;
end do: # R. J. Mathar, Oct 02 2018
MATHEMATICA
A091113 = Select[4 Range[0, 10^5] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^5] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091113, Between[{A091236[[i]], A091236[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091236] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 5}] (* Robert Price, May 30 2019 *)
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Enoch Haga, Mar 30 2004
EXTENSIONS
a(9)-a(13) from Bert Dobbelaere, Dec 19 2018
STATUS
approved