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.

A093397 Absolute value of difference between counts of uninterrupted runs of single nonprimes in A093183 and A093184.

Original entry on oeis.org

0, 1, 2, 6, 9, 31, 88, 205, 1636, 2884, 4569, 30270, 14874
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=2 because in A093183 the count is 74 and in A093184 the count is 76. 76-74=2.

Crossrefs

Cf. A093183 A093184 A092636 A092637 A092638.

Formula

Take the absolute value of differences between counts of runs of single nonprimes congruent to 1 mod 4 and 3 mod 4

Extensions

a(9)-a(13) from Robert Price, May 30 2019

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

A093188 Number of consecutive runs of 3 odd nonprimes congruent to 3 mod 4 below 10^n.

Original entry on oeis.org

0, 0, 5, 49, 356, 2678, 21085, 166814, 1345812, 11080939, 92699035, 786630700, 6757485506
Offset: 1

Views

Author

Enoch Haga, Mar 30 2004

Keywords

Examples

			a(3)=5 because 5 nonprime runs of 3 occur below 10^3, each run interrupted by a nonprime congruent to 1 mod 4.

Crossrefs

Cf. A093183, A093184, A093185, A093186, A093187, A093397, A093398, A093399, A092643.

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]]}]]] == 3, AppendTo[lst, Last[s]]], {i, Length[A091113] - 1}]; Table[Count[lst, x_ /; x < 10^n], {n, 4}]  (* Robert Price, May 31 2019 *)

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 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

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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