A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.

0, 3, 74, 1114, 13437, 151311, 1642197, 17405273, 181925434, 1883327626, 19364371468, 198115934511, 2019328584101

Offset: 1

Author

Enoch Haga, Mar 30 2004

Keywords

• more
• nonn

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

Examples

			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
		

Crossrefs

Cf. A091113, A091236, A093184-A093188, A093397-A093399, A092636.

Programs

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 *)

Extensions

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018