cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A093399 Absolute value of difference between counts of uninterrupted runs of 3 nonprimes in A093187 and A093188.

Original entry on oeis.org

0, 1, 5, 4, 34, 116, 130, 241, 2187, 3076, 9683, 33067, 133346
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=5 because in A093187 the count is 10 and in A093188 the count is 5. 10-5=5.

Crossrefs

Cf. A093187 A093188 A092642 A092643 A092644.

Formula

Take the absolute value of differences between counts of runs of 3 odd nonprimes congruent to 1 mod 4 and 3 mod 4
a(n) = abs(A093187(n) - A093188(n))

Extensions

a(9)-a(13) from Lucas A. Brown, Sep 19 2024

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

Original entry on oeis.org

0, 3, 74, 1114, 13437, 151311, 1642197, 17405273, 181925434, 1883327626, 19364371468, 198115934511, 2019328584101
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

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

A093184 Number of consecutive runs of just 1 odd nonprime congruent to 3 mod 4 below 10^n.

Original entry on oeis.org

0, 4, 76, 1120, 13428, 151342, 1642285, 17405478, 181923798, 1883330510, 19364376037, 198115964781, 2019328569227
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=76 because 76 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4

Crossrefs

Cf. A093183, A093185, A093186, A093187, A093188, A093397, A093398, A093399, A092637.

Programs

  • Mathematica
    A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091236, Between[{A091113[[i]], A091113[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091113] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 30 2019 *)

Formula

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 3 mod 4 occurs before interruption of a nonprime congruent to 1 mod 4.

Extensions

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

A093185 Number of consecutive runs of 2 odd nonprimes congruent to 1 mod 4 below 10^n.

Original entry on oeis.org

1, 4, 33, 309, 2805, 25566, 230989, 2106529, 19303539, 177948527, 1649241049, 15360074924, 143682925080
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=33 because 33 nonprime runs of 2 occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4

Crossrefs

Cf. A093183 A093184 A093186 A093187 A093187 A093188 A093397 A093398 A093399 A092639.

Programs

  • Mathematica
    A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &];
A091236 = Join[{0}, Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &]];
lst = {}; Do[If[Length[s = Select[A091113, Between[{A091236[[i]], A091236[[i + 1]]}]]] == 2, AppendTo[lst, Last[s]]], {i, Length[A091236] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 30 2019 *)

Formula

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if 2 nonprimes congruent to 1 mod 4 occur before interruption of a nonprime congruent to 3 mod 4.

Extensions

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

A093186 Number of consecutive runs of 2 odd nonprimes congruent to 3 mod 4 below 10^n.

Original entry on oeis.org

0, 4, 36, 307, 2848, 25651, 231031, 2106565, 19307362, 177948719, 1649246163, 15360077721, 143683073300
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=36 because 36 nonprime runs of 2 occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4.

Crossrefs

Cf. A093183 A093184 A093185 A093187 A093188 A093397 A093398 A093399 A092640.

Programs

  • Mathematica
    A091113 = Select[4 Range[0, 10^4] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^4] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091236, Between[{A091113[[i]], A091113[[i + 1]]}]]] == 2, AppendTo[lst, Last[s]]], {i, Length[A091113] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 30 2019 *)

Formula

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if 2 nonprimes congruent to 3 mod 4 occur before interruption of a nonprime congruent to 1 mod 4

Extensions

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

A093187 Number of consecutive runs of 3 odd nonprimes congruent to 1 mod 4 below 10^n.

Original entry on oeis.org

0, 1, 10, 53, 390, 2794, 21215, 167055, 1347999, 11084015, 92708718, 786663767, 6757618852
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=10 because 10 nonprime runs of 3 occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4.

Crossrefs

Cf. A092642, A093183, A093184, A093185, A093186, A093188, A093397, A093398, A093399.

Programs

  • Mathematica
    Accumulate@ Table[Length@ SequencePosition[Range[10^n + 1, 10^(n + 1) - 1, 2] /. {p_ /; PrimeQ@ p -> Nothing, k_ /; Mod[k, 4] == 1 -> 1, k_ /; Mod[k, 4] == 3 -> 3}, {1, 1, 1, 3}], {n, 0, 6}] (* Michael De Vlieger, Jan 02 2017 *)

Formula

Generate the odd nonprime sequence with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if 3 nonprimes congruent to 1 mod 4 occur before interruption of a nonprime congruent to 3 mod 4.

Extensions

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018
Showing 1-6 of 6 results.

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