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 1-4 of 4 results.

A145986 n-th prime in the first occurrence of at least n consecutive primes of the form 4k + 1.

Original entry on oeis.org

5, 17, 101, 409, 2633, 11657, 11677, 11681, 11689, 373777, 766373, 3358373, 12205121, 12270281, 12270301, 12270317, 297388097, 297779509, 297779513, 1113443473, 1113443521, 1113443533, 1113443549, 1113443561, 84676453373, 84676453429
Offset: 1

Views

Author

Enoch Haga, Oct 26 2008

Keywords

Comments

a(1)=5 is same as A055623(1) because 5 is a single-digit number.

Examples

			a(2)=17 because this is the 2nd prime in the first run of 2 primes where p == 1 mod 4.

References

  • Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007, pp. 30-31. ISBN 978-1-885794-24-6

Crossrefs

Cf. A055623, A054624, A145988, A145989, A145990, A145991, A145992, A145993, A145994.

Programs

  • Mathematica
    Flatten[Table[SequencePosition[Table[If[Mod[p-1,4]==0,1,0],{p,Prime[Range[250000]]}],PadRight[ {},n,1],1],{n,12}],1][[;;,-1]]//Prime (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jul 14 2024 *)
  • PARI
    r=0;c=0;forprime(p=2,4e9,if(p%4==1,if(c++>r,r=c;print1(p", ")),c=0)) \\ Charles R Greathouse IV, Mar 22 2011
  • UBASIC
    10 'cluster primes
20 C=1:input "end #";L
40 for N=3 to L step 2
50 S=int(sqrt(N))
60 for A=3 to S step 2
70 B=N/A
80 if int(B)*A=N then cancel for:goto 170
90 next A
100 C=C+1: E=int(N/4):R=N-(4*E)
120 if R=1 then print N;:C1=C1+1:T1=T1+1:print T1
130 if R=3 then T1=0:print " ";N;:C3=C3+1:T2=T2+1:print T2
150 if R=1 then T2=0
160 if T1>10 or T2>10 then stop
170 next
180 print "Total primes=";C;:print "Type A:";C1;" Type B:";C3

Extensions

Entry rewritten and a(13)-a(26) added by Charles R Greathouse IV, Mar 22 2011
Edited by M. F. Hasler, May 02 2015
Definition clarified by N. J. A. Sloane, Dec 18 2022

A145994 Last prime in a run of at least 2 consecutive primes of the form 4k+3.

Original entry on oeis.org

11, 23, 47, 71, 83, 107, 131, 167, 227, 311, 367, 383, 443, 503, 631, 647, 691, 727, 751, 827, 863, 919, 971, 991, 1091, 1171, 1283, 1319, 1427, 1451, 1471, 1487, 1543, 1583, 1667, 1787, 1847, 1871, 1987, 2011, 2087, 2111, 2207, 2267, 2351, 2411, 2467, 2543, 2591, 2671, 2687
Offset: 1

Views

Author

Enoch Haga, Oct 26 2008

Keywords

Examples

			a(1)=11 because this sequence includes consecutive runs of any length >1 and this ending term in a run of 2 is 11.

References

  • Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007. Pp. 30-31. ISBN 978-1-885794-24-6

Crossrefs

Cf. A039702, A055623, A145986, A145988, A145990, A145991, A145992 (run lengths), A145993 (first prime in run)

Programs

  • Maple
    A145994 := proc()
    local m,p,r,i,lp ;
    m := 3 ;
    p := 2 ;
    r := 0 ;
    for i from 2 to 1000 do
        if modp(p,4) = m then
            r := r+1 ;
        else
            if r > 1 then
                printf("%d,",prevprime(p)) ;
            end if;
            r := 0;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145994() ; # R. J. Mathar, Aug 29 2018
  • Mathematica
    Last /@ Select[Split[Select[4Range[1000]+3, PrimeQ], #2 == NextPrime[#1]&], Length[#]>1&] (* Jean-François Alcover, Mar 26 2020 *)
  • UBASIC
    10 'cluster primes
20 C=1
30 input "end #";L
40 for N=3 to L step 2
50 S=int(sqrt(N))
60 for A=3 to S step 2
70 B=N/A
80 if int(B)*A=N then cancel for:goto 170
90 next A
100 C=C+1
110 E=N/4:E=int(E):R=N-(4*E)
120 if R=1 then print N;:C1=C1+1:T1=T1+1:print T1
130 if R=3 then T1=0
140 if R=3 then print " ";N;:C3=C3+1:T2=T2+1:print T2
150 if R=1 then T2=0
160 if T1>10 or T2>10 then stop
170 next
180 print "Total primes=";C;:print "Type A";C1;"Type B";C3

A145988 Ending prime: n-th prime in the first occurrence of n consecutive primes of the form 4k + 3.

Original entry on oeis.org

3, 11, 223, 227, 491, 499, 503, 36607, 39703, 183283, 241727, 241739, 241771, 9177607, 9177611, 95949631, 105639463, 341118731, 727335359, 727335379, 1786054619, 1786054631, 22964264759, 54870713999, 79263248759
Offset: 1

Views

Author

Enoch Haga, Oct 26 2008

Keywords

Comments

a(1)=3 is the same as A055624(1) because 3 is a single-digit number.

Examples

			a(2)=11 because this is the 2nd prime in the first run of 2 primes where p == 3 mod 4.

References

  • Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007, pp. 30-31. ISBN 978-1-885794-24-6

Crossrefs

Cf. A055623, A054624, A145986, A145989, A145990, A145991, A145992, A145993, A145994.

Programs

  • Mathematica
    Prime[#]&/@Flatten[Table[SequencePosition[If[Mod[#,4]==3,1,0]&/@Prime[ Range[ 615000]],PadRight[{},n,1],1],{n,15}],1][[All,2]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Jun 17 2022 *)
  • PARI
    r=0; c=0; forprime(p=2, 4e9, if(p%4==3, if(c++>r, r=c; print1(p", ")), c=0)) \\ Charles R Greathouse IV, Mar 22 2011
  • UBASIC
    10 'cluster primes 20 C=1 30 input "end #";L 40 for N=3 to L step 2 50 S=int(sqrt(N)) 60 for A=3 to S step 2 70 B=N/A 80 if int(B)*A=N then cancel for:goto 170 90 next A 100 C=C+1 110 E=N/4:E=int(E):R=N-(4*E) 120 if R=1 then print N;:C1=C1+1:T1=T1+1:print T1 130 if R=3 then T1=0 140 if R=3 then print " ";N;:C3=C3+1:T2=T2+1:print T2 150 if R=1 then T2=0 160 if T1>10 or T2>10 then stop 170 next 180 print "Total primes=";C;:print "Type A";C1;"Type B";C3

Extensions

Entry rewritten by, and a(14)-a(25) from, Charles R Greathouse IV, Mar 22 2011

A145993 Primes that start a run of at least 2 consecutive primes of the form 4k+3.

Original entry on oeis.org

7, 19, 43, 67, 79, 103, 127, 163, 199, 307, 359, 379, 439, 463, 619, 643, 683, 719, 739, 823, 859, 883, 967, 983, 1087, 1163, 1279, 1303, 1423, 1439, 1459, 1483, 1499, 1559, 1663, 1783, 1811, 1867, 1979, 1999, 2083, 2099, 2179, 2239, 2347, 2399, 2447, 2531, 2579, 2659, 2683, 2699, 2803, 2843, 2879
Offset: 1

Views

Author

Enoch Haga, Oct 26 2008

Keywords

Examples

			a(1)=7 because this sequence includes consecutive runs of any length and this first term >1 in a run of 2 is 7.

References

  • Enoch Haga, Exploring Primes on Your PC and the Internet, 1994-2007. Pp. 30-31. ISBN 978-1-885794-24-6

Crossrefs

Cf. A039702, A055623, A145986, A145988, A145989, A145990, A145991, A145992 (run lengths) A145994.

Programs

  • Maple
    A145993 := proc()
    local m,p,r,i,sp ;
    m := 3 ;
    p := 2 ;
    r := 0 ;
    sp := -1 ;
    for i from 2 to 1000 do
        if modp(p,4) = m then
            r := r+1 ;
            if r = 1 then
                sp := p ;
            end if;
        else
            if r > 1 then
                printf("%d,",sp) ;
            end if;
            r := 0;
            sp := -1 ;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145993() ; # R. J. Mathar, Aug 29 2018
  • Mathematica
    Most[First /@ Select[ SplitBy[ Prime@ Range@ 425, Mod[#, 4] &], Mod[#[[1]], 4] == 3 && Length[#] > 1 &]] (* Giovanni Resta, Aug 29 2018 *)

Extensions

619 inserted by R. J. Mathar, Aug 29 2018
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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