A071698 -id:A071698 - cp's OEIS frontend

A071695 Lesser members of twin prime pairs of form (4k+1, 4k+3), k > 0.

5, 17, 29, 41, 101, 137, 149, 197, 269, 281, 461, 521, 569, 617, 641, 809, 821, 857, 881, 1049, 1061, 1229, 1277, 1289, 1301, 1481, 1697, 1721, 1877, 1949, 1997, 2081, 2129, 2141, 2237, 2309, 2381, 2549, 2657, 2729, 2789, 2801, 2969

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding greater members: A071696(n).
Or, lesser members of twin prime pairs (A001359) which are also Pythagorean primes (A002144). Intersection of A001359 and A002144. - Zak Seidov, Apr 25 2008
A010051(a(n)) * A010051(a(n)+2) = 1. - Reinhard Zumkeller, Nov 10 2013

Zak Seidov, Table of n, a(n) for n = 1..1000.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A001359, A002144, A071698, A071697.

a071695 n = a071695_list !! (n-1)
a071695_list = [p | p <- a002144_list, a010051' (p + 2) == 1]
-- Reinhard Zumkeller, Nov 10 2013

Select[ Prime@ Range@ 1000, Mod[#, 4] == 1 && PrimeQ[# + 2] &] (* Robert G. Wilson v, Jan 22 2012 *)

p=2;forprime(q=3,1e4,if(q-p==2 && p%4==1, print1(p", "));p=q) \\ Charles R Greathouse IV, Mar 20 2013

A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.

Original entry on oeis.org

15, 143, 3599, 5183, 11663, 32399, 36863, 51983, 57599, 97343, 121103, 176399, 186623, 359999, 435599, 685583, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

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

Subsequence of A037074, A071700 and of A182140.

Cf. A071697, a(n) = A071698(n)*A071699(n).

a071700 n = a071700_list !! (n-1)
a071700_list = [x * y | x <- [3, 7 ..], a010051' x == 1,
                        let y = x + 2, a010051' y == 1]
-- Reinhard Zumkeller, Aug 05 2014

for(k=0,1e3,if(isprime(4*k+3)&&isprime(4*k+5),print1(16*k^2+32*k +15", "))) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(t); n%16==15 && issquare(n+1,&t) && isprime(t-1) && isprime(t+1) \\ Charles R Greathouse IV, Dec 12 2016

list(lim)=my(v=List(),p=3); forprime(q=5,sqrtint(1+lim\1)+1, if(q-p==2 && p%4==3, listput(v,p*q)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 12 2016

a(n) >> n^2 log^4 n. - Charles R Greathouse IV, Jul 03 2013

A071699 Greater members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

Original entry on oeis.org

5, 13, 61, 73, 109, 181, 193, 229, 241, 313, 349, 421, 433, 601, 661, 829, 1021, 1033, 1093, 1153, 1321, 1429, 1453, 1489, 1609, 1621, 1669, 1789, 1873, 1933, 2029, 2089, 2113, 2269, 2341, 2593, 2689, 2713, 3001, 3121, 3169, 3253, 3301

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding lesser members: A071698(n).

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

Cf. A071696, A071698, A071700.
Cf. A010051, subsequence of A016813.
Intersection of A006512 and A002144.

a071699 n = a071699_list !! (n-1)
a071699_list = [x | x <- [5, 9 ..], a010051' x == 1, a010051' (x-2) == 1]
-- Reinhard Zumkeller, Aug 05 2014

[4*(k+1)+1:k in [0..1000]|IsPrime(4*k+3) and IsPrime(4*k+5)]; // Marius A. Burtea, Nov 06 2019

#+2&/@Select[4Range[0,850]+3,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = 2*A241557(n+1) + 1. - Hilko Koning, Nov 06 2019

A072029 Swap twin prime pairs of form (4k+3,4(k+1)+1) in prime factorization of n.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 7, 8, 25, 6, 13, 20, 11, 14, 15, 16, 17, 50, 19, 12, 35, 26, 23, 40, 9, 22, 125, 28, 29, 30, 31, 32, 65, 34, 21, 100, 37, 38, 55, 24, 41, 70, 43, 52, 75, 46, 47, 80, 49, 18, 85, 44, 53, 250, 39, 56, 95, 58, 61, 60, 59, 62, 175, 64, 33

Offset: 1

Reinhard Zumkeller, Jun 07 2002

			a(42) = a(2*3*7) = a(2)*a(3)*a(7) = a(2)*a(4*0+3)*a(7) = 2*(4*1+1)*7 = 2*5*7 = 70.

Cf. A061898, A064505, A071698, A071699, A072026, A072027, A072028.

a[n_] := Product[{p, e} = pe; Which[
     Mod[p, 4] == 3 && PrimeQ[p + 2], p + 2,
     Mod[p, 4] == 1 && PrimeQ[p - 2], p - 2,
     True, p]^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 21 2021 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; if(p == 2, p, if(p%4 == 3 && isprime(p+2), p+2, if(p%4 == 1 && isprime(p-2), p-2, p)))^f[i,2]);} \\ Amiram Eldar, Feb 26 2024

Multiplicative with a(p) = (if p mod 4 = 3 and p+2 is prime then p+2 else (if p mod 4 = 1 and p-2 is prime then p-2 else p)), p prime.
a(a(n))=n, a self-inverse permutation of natural numbers.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{(p < q) swapped pair} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 1.37140598833326962... . - Amiram Eldar, Feb 26 2024

A071702 Number of twin prime pairs <= n of form (4k+3,4(k+1)+1), k>=0.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

As for A071538 the convention is followed that a twin prime pair is <= n if its smaller member is <= n.

If {a,b} is a twin prime pair (aHarvey P. Dale, Nov 12 2021

			a(60)=3, since (59,61) is included along with (3,5) and (11,13).

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

a(n) = A071538(n) - A071701(n), cf. A071698, A071699.

Accumulate[Table[If[AllTrue[{n,n+2},PrimeQ]&&Mod[n,4]==3,1,0],{n,100}]] (* Harvey P. Dale, Nov 12 2021 *)

A175606 Primes p of the form 4*k+3 such that p+2 is prime and p-1 is nonsquarefree.

Original entry on oeis.org

1151, 1451, 1667, 3251, 3851, 4019, 5651, 6359, 6551, 6959, 7547, 11351, 11831, 12251, 13691, 15731, 15887, 16451, 17987, 18131, 18251, 19751, 20231, 22091, 26951, 27539, 28751, 30851, 31151, 32831, 35051, 37571, 38651, 38711, 40151, 43319, 44279, 44771, 45179

Offset: 1

Zak Seidov, Jul 22 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A071698.

Cf. A089188.

Select[Range[3,99299,4],!SquareFreeQ[ #-1]&&PrimeQ[ # ]&&PrimeQ[ #+2]&]

Name corrected by Amiram Eldar, Apr 23 2022

A267573 a(n) = prime(n) + (prime(n) mod 4).

Original entry on oeis.org

4, 6, 6, 10, 14, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 62, 62, 70, 74, 74, 82, 86, 90, 98, 102, 106, 110, 110, 114, 130, 134, 138, 142, 150, 154, 158, 166, 170, 174, 182, 182, 194, 194, 198, 202, 214, 226, 230, 230, 234, 242, 242, 254, 258, 266, 270

Offset: 1

Emre APARI, Jan 17 2016

The primes corresponding to the cases where a(n) = a(n+1) can be found in A071698. - Michel Marcus, Jan 17 2016

			p=19; 19 + (19 modulo 4) = 22.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A039702, A071698, A083220.

[NthPrime(n)+(NthPrime(n) mod 4): n in [1..100]]; // Vincenzo Librandi, Jan 17 2016

A267573:=n->ithprime(n)+(ithprime(n) mod 4): seq(A267573(n), n=1..100); # Wesley Ivan Hurt, Jan 17 2016

Table[Prime[n] + Mod[Prime[n], 4], {n, 60}] (* Vincenzo Librandi, Jan 17 2016 *)
#+Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Jun 12 2020 *)

a(n) = prime(n) + (prime(n) % 4); \\ Michel Marcus, Jan 17 2016

lista(nn) = forprime(p=2, nn, print1(p + p % 4, ", ")); \\ Altug Alkan, Jan 17 2016

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

a(n) = A083220(prime(n)). - Michel Marcus, Jan 17 2016

More terms from Vincenzo Librandi, Jan 17 2016

A074381 (p-1)/2 mod 2, where p is the n-th prime for which p+2 is also prime; i.e., a(n)=0 if p==1 (mod 4), a(n)=1 if p==3 (mod 4).

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Sep 24 2002

Cf. A001359, A071695, A071698.

Mod[(Select[Prime/@Range[600], PrimeQ[ #+2]&]-1)/2, 2]

Edited by Dean Hickerson and Robert G. Wilson v, Oct 09 2002

A074395 A 7-way classification of the primes.

Original entry on oeis.org

6, 1, 0, 5, 1, 4, 0, 5, 3, 0, 3, 4, 0, 5, 3, 2, 1, 2, 5, 1, 2, 5, 3, 4, 4, 0, 5, 1, 4, 2, 5, 3, 0, 3, 0, 3, 2, 5, 3, 2, 1, 2, 1, 4, 0, 5, 5, 5, 1, 4, 2, 1, 2, 3, 2, 3, 0, 3, 4, 0, 3, 2, 5, 1, 4, 2, 3, 2, 1, 4, 2, 5, 3, 2, 5, 3, 4, 4, 4, 2, 1, 2, 1, 2, 5, 3, 4, 4, 0, 5, 5, 5, 5, 5, 5, 3, 4, 0, 3, 2, 3, 2, 3, 0, 3

Offset: 1

Roger L. Bagula, Sep 24 2002

nonn

There are seven types of consecutive primes modulus 4 and whether or not they are twin primes. They are a (1, 3, paired), (3, 1, paired), (1, 3, not paired), (3, 1, not paired), (1, 1), (3, 3) and p(m)=2. Each case is mapped to a number from zero to six, respectively. Here the word paired means that the consecutive primes are twins.

The initial digit (6) occurs but once and the frequency for the digits 0 and 1 decreased with added terms.

Cf. A071696, A071698.

a = {}; Do[p = Prime[n]; q = Prime[n + 1]; a = Append[a, Which[ Mod[p, 4] == 1 && Mod[q, 4] == 3 && p + 2 == q, 0, Mod[p, 4] == 3 && Mod[q, 4] == 1 && p + 2 == q, 1, Mod[p, 4] == 1 && Mod[q, 4] == 3 && p + 2 != q, 2, Mod[p, 4] == 3 && Mod[q, 4] == 1 && p + 2 != q, 3, Mod[p, 4] == 1 && Mod[q, 4] == 1, 4, Mod[p, 4] == 3 && Mod[q, 4] == 3, 5, p == 2, 6]]; p = q, {n, 1, 105}]; a

Edited and extended by Robert G. Wilson v, Oct 03 2002

cp's OEIS Frontend

A071695 Lesser members of twin prime pairs of form (4*k+1, 4*k+3), k > 0.

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A071700 Product of twin primes of form (4*k+3,4*(k+1)+1), k>=0.

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A071699 Greater members of twin prime pairs of form (4*k+3, 4*k+5), k >= 0.

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A072029 Swap twin prime pairs of form (4*k+3,4*(k+1)+1) in prime factorization of n.

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A071702 Number of twin prime pairs <= n of form (4*k+3,4*(k+1)+1), k>=0.

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A175606 Primes p of the form 4*k+3 such that p+2 is prime and p-1 is nonsquarefree.

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A267573 a(n) = prime(n) + (prime(n) mod 4).

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A071695 Lesser members of twin prime pairs of form (4k+1, 4k+3), k > 0.

A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.

A071699 Greater members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

A072029 Swap twin prime pairs of form (4k+3,4(k+1)+1) in prime factorization of n.

A071702 Number of twin prime pairs <= n of form (4k+3,4(k+1)+1), k>=0.