A039702 -id:A039702 - cp's OEIS frontend

A100672 a(1) = 1; thereafter, a(n) = 1 if n-th prime is 3 mod 4, 0 if n-th prime is 1 mod 4.

1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 06 2004

Second least-significant bit in the binary expansion of the n-th prime.

a(n)=1 iff prime(n) is a member of A045326 (equivalently for n>1, iff prime(n)-3 is divisible by 4).

			a(2)=1 because prime(2)=11_2 (in binary; decimal = 3_10) and its 2^1 bit is 1.
a(3)=0 because prime(3)=101_2 (in binary; decimal = 5_10) and its 2^1 bit is 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A000040, A045326, A002144, A002145, A039702.

RUNS transform is essentially A091237.

A100672 := proc(n)
        if n = 1 then
                1 ;
        else
                ((ithprime(n) mod 4)-1)/2;
        end if;
end proc: # R. J. Mathar, Oct 06 2011

Table[Reverse[RealDigits[Prime[k], 2][[1]]][[2]], {k, 1, 128}]

for(k=1,105,print1( bittest(prime(k), 1), ", ")) \\ Washington Bomfim, Jan 18 2011

from sympy import prime
def A100672(n): return int(prime(n)>>1&1) # Chai Wah Wu, Jun 23 2023

a(n) = 1-A098033(n), n>1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008

a(n) = floor(prime(n)/2) mod 2. - Alois P. Heinz, Jul 16 2024

Edxited by N. J. A. Sloane, Jan 11 2025

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

Enoch Haga, Oct 26 2008

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

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

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

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

Last /@ Select[Split[Select[4Range[1000]+3, PrimeQ], #2 == NextPrime[#1]&], Length[#]>1&] (* Jean-François Alcover, Mar 26 2020 *)

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

A145990 Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).

Original entry on oeis.org

13, 37, 89, 109, 193, 229, 277, 313, 349, 389, 449, 509, 613, 661, 701, 757, 797, 853, 877, 929, 997, 1093, 1109, 1193, 1237, 1297, 1373, 1429, 1489, 1549, 1597, 1609, 1637, 1669, 1709, 1733, 1789, 1873, 1889, 1933, 1993, 2069, 2113, 2137, 2153, 2213, 2269

Offset: 1

Enoch Haga, Oct 26 2008

			a(1)=13 because this sequence includes consecutive runs of any length and this first term > 1 in a run of 2 is 13.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A039702, A055623, A054624, A145986, A145988 - A145994.

for i from 2 to 300 do
        if (ithprime(i) mod 4) = 1  and ithprime(i-1) mod 4 <> 1 and ithprime(i+1) mod 4 = 1 then
                printf("%d,",ithprime(i)) ;
        end if;
end do: # R. J. Mathar, Sep 30 2011

Prime[#+1]&/@(SequencePosition[Table[If[Mod[n,4]==1,1,0],{n,Prime[ Range[ 350]]}],{0,1,1},Overlaps->False][[All,1]]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)

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

Corrected and extended by Harvey P. Dale, Aug 02 2017

A075520 4*prime(n) + (prime(n) mod 4).

Original entry on oeis.org

10, 15, 21, 31, 47, 53, 69, 79, 95, 117, 127, 149, 165, 175, 191, 213, 239, 245, 271, 287, 293, 319, 335, 357, 389, 405, 415, 431, 437, 453, 511, 527, 549, 559, 597, 607, 629, 655, 671, 693, 719, 725, 767, 773, 789, 799, 847, 895, 911, 917, 933, 959, 965

Offset: 1

Reinhard Zumkeller, Sep 19 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A039702.

a075520 n = a075520_list !! (n-1)
a075520_list = zipWith (+) a001749_list a039702_list
-- Reinhard Zumkeller, Feb 20 2012

4#+Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Mar 10 2016 *)

a(n) = A001749(n) + A039702(n).

A145989 Run lengths of consecutive primes == 1 (mod 4) where the run length is at least 2.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 3, 2, 4, 2, 2, 2, 3, 3, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Enoch Haga, Oct 26 2008

The run lengths of 1's in A039702 are 1, 2, 1, 2, 1, 1, 1, 3,.. as listed in A091318. Deleting all 1's from A091318 generates this sequence here. - R. J. mathar, Sep 30 2011

The maximum run length in the first 1000 terms is 9. - Harvey P. Dale, Jul 27 2025

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

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055623, A054624, A145986, A145988 - A145994.

Length/@Select[Split[Table[If[Mod[p,4]==1,1,0],{p,Prime[Range[500]]}]],#[[1]]==1&&Length[#]>1&] (* Harvey P. Dale, Jul 27 2025 *)

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

A145992 Run lengths of 2 or more consecutive primes of the form 4k+3.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 7, 2, 2, 2, 2, 3, 2, 2, 5, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 5, 5, 2, 2, 4, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 3, 3, 2, 3, 4, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3

Offset: 1

Enoch Haga, Oct 26 2008

			a(1) = 2 counts the two 3's from A039702(4) to A039702(5).
a(9) = 4 counts the four 3's from A039702(46) to A039702(49).
a(14)= 7 counts the seven 4's from A039702(90) to A039702(96).

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A039702, A055623, A145986, A145988, A145989, A145990, A145991, A145993, A145994.

A145992 := proc()
    local m,p,r,i ;
    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,",r) ;
            end if;
            r := 0;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
A145992() ; # R. J. Mathar, Aug 29 2018

Most[Length /@ Select[ SplitBy[ Prime@ Range@ 780, Mod[#, 4] &], Mod[#[[1]], 4] == 3 && Length[#] > 1 &]] (* Giovanni Resta, Aug 29 2018 *)
Length/@Select[Split[Table[If[Mod[n,4]==3,1,0],{n,Prime[Range[ 1000]]}]], FreeQ[ #,0]&]/.(1->Nothing) (* Harvey P. Dale, Jul 27 2020 *)

Corrected by R. J. Mathar, Aug 29 2018

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

Enoch Haga, Oct 26 2008

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

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

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

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

Most[First /@ Select[ SplitBy[ Prime@ Range@ 425, Mod[#, 4] &], Mod[#[[1]], 4] == 3 && Length[#] > 1 &]] (* Giovanni Resta, Aug 29 2018 *)

619 inserted by R. J. Mathar, Aug 29 2018

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

Original entry on oeis.org

2, 3, 3, 1, 5, 5, 3, 1, 7, 7, 3, 5, 1, 11, 11, 1, 1, 3, 13, 13, 13, 1, 5, 7, 3, 17, 17, 17, 3, 1, 9, 5, 19, 19, 19, 19, 3, 5, 3, 9, 1, 23, 23, 23, 23, 1, 5, 9, 1, 7, 3, 29, 29, 29, 29, 3, 1, 1, 3, 9, 5, 31, 31, 31, 31, 31, 1, 1, 7, 9, 15, 11, 3, 37, 37, 37, 37, 37, 1, 5, 1, 13, 19, 15, 7, 3, 41

Offset: 1

Reinhard Zumkeller, Jan 19 2003

The right border of the triangle are the primes: T(n,n)=A000040(n); T(n,1)=A039702(n), T(n,2)=A039704(n) for n>1, T(n,3)=A007652(n) for n>2, T(n,4)=A039712(n) for n>3;

			Triangle begins:
  2;
  3, 3;
  1, 5, 5;
  3, 1, 7,  7;
  3, 5, 1, 11, 11;
  1, 1, 3, 13, 13, 13;
  1, 5, 7,  3, 17, 17, 17;
  ...

Cf. A001747, A079951, A079952, A079953.

A079950 := proc(n,k)
    modp(ithprime(n),2*ithprime(k)) ;
end proc:
seq(seq(A079950(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 28 2017

T(n,k) = prime(n) % (2*prime(k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 21 2017

T(n, k) = prime(n) mod 2*prime(k), 1<=k<=n.

A082542 a(n) = prime(n) + 2 - (prime(n) mod 4).

Original entry on oeis.org

2, 2, 6, 6, 10, 14, 18, 18, 22, 30, 30, 38, 42, 42, 46, 54, 58, 62, 66, 70, 74, 78, 82, 90, 98, 102, 102, 106, 110, 114, 126, 130, 138, 138, 150, 150, 158, 162, 166, 174, 178, 182, 190, 194, 198, 198, 210, 222, 226, 230, 234, 238, 242, 250, 258, 262, 270, 270, 278

Offset: 1

Reinhard Zumkeller, May 02 2003

For k > 1: a(k+1) = a(k) if and only if prime(k) == 1 modulo 4 and prime(k+1) = prime(k) + 2, see A071695 and A071696.

			a(2) = 2 because the second prime is 3, and 3 + 2 - 3 = 2.
a(3) = 6 because the third prime is 5, and 5 + 2 - 1 = 6.
a(4) = 6 because the fourth prime is 7, and 7 + 2 - 3 = 6.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060294, A070750, A076342.

Cf. A039702, A071695, A071696, A100484.

[2 + NthPrime(n) - (NthPrime(n) mod 4): n in [1..60]]; // G. C. Greubel, Nov 14 2018

Table[Prime[n] + 2 - Mod[Prime[n], 4], {n, 60}] (* Alonso del Arte, Feb 23 2015 *)
#+2-Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Aug 24 2025 *)

vector(60, n, 2 + prime(n) - lift(Mod(prime(n),4))) \\ G. C. Greubel, Nov 14 2018

a(n) = A000040(n) + A070750(n).
a(n+1) = p + (-1/p) = p + (-1)^((p-1)/2), where p is the n-th odd prime and (-1/p) denotes the value of Legendre symbol. - Lekraj Beedassy, Mar 17 2005
a(n) = (A000040(n) OR 3) - 1. - Jon Maiga, Nov 14 2018
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A100484(n) - A076342(n).
Product_{n>=1} a(n)/prime(n) = 2/Pi (A060294). (End)

A075518 a(n) = floor(prime(n)/4).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 4, 5, 7, 7, 9, 10, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 25, 25, 26, 27, 28, 31, 32, 34, 34, 37, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 49, 52, 55, 56, 57, 58, 59, 60, 62, 64, 65, 67, 67, 69, 70, 70, 73, 76, 77, 78, 79, 82, 84, 86, 87, 88, 89, 91

Offset: 1

Reinhard Zumkeller, Sep 19 2002

a(n)*4 + A039702(n) = A000040(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A075519.

a075518 = (`div` 4) . a000040  -- Reinhard Zumkeller, Oct 10 2013

Floor[Prime[Range[80]]/4] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = prime(n) \ 4; \\ Joerg Arndt, Aug 04 2013

cp's OEIS Frontend

A100672 a(1) = 1; thereafter, a(n) = 1 if n-th prime is 3 mod 4, 0 if n-th prime is 1 mod 4.

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

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UBASIC

A145990 Primes which start a run of at least length 2 of consecutive primes == 1 (mod 4).

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UBASIC

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A075520 4*prime(n) + (prime(n) mod 4).

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A145989 Run lengths of consecutive primes == 1 (mod 4) where the run length is at least 2.

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Mathematica

UBASIC

A145992 Run lengths of 2 or more consecutive primes of the form 4k+3.

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Programs

Maple

Mathematica

Extensions

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

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