A071699 -id:A071699 - cp's OEIS frontend

A071696 Greater members of twin prime pairs of form (4k+1,4k+3), k>0.

7, 19, 31, 43, 103, 139, 151, 199, 271, 283, 463, 523, 571, 619, 643, 811, 823, 859, 883, 1051, 1063, 1231, 1279, 1291, 1303, 1483, 1699, 1723, 1879, 1951, 1999, 2083, 2131, 2143, 2239, 2311, 2383, 2551, 2659, 2731, 2791, 2803, 2971

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding lesser members: A071695(n).

A010051(a(n)) * A010051(a(n)-2) = 1. - Reinhard Zumkeller, Nov 10 2013

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

Cf. A006512, A071699, A071697.

Cf. Subsequence of A002145.

a071696 n = a071696_list !! (n-1)
a071696_list = [p | p <- tail a002145_list, a010051' (p - 2) == 1]
-- Reinhard Zumkeller, Nov 10 2013

Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[ (#[[1]]-1)/4]&][[All,2]] (* Harvey P. Dale, Aug 27 2021 *)
Select[Table[4k+{1,3},{k,750}],AllTrue[#,PrimeQ]&][[;;,2]] (* Harvey P. Dale, Sep 10 2024 *)

A071698 Lesser members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

Original entry on oeis.org

3, 11, 59, 71, 107, 179, 191, 227, 239, 311, 347, 419, 431, 599, 659, 827, 1019, 1031, 1091, 1151, 1319, 1427, 1451, 1487, 1607, 1619, 1667, 1787, 1871, 1931, 2027, 2087, 2111, 2267, 2339, 2591, 2687, 2711, 2999, 3119, 3167, 3251, 3299

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding greater members: A071699(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A001359, A071695, A071700.

Cf. A010051, subsequence of A004767.

a071698 n = a071698_list !! (n-1)
a071698_list = [x | x <- [3, 7 ..], a010051' x == 1, a010051' (x+2) == 1]
-- Reinhard Zumkeller, Aug 05 2014

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

Transpose[Select[Table[4n+{3,5},{n,0,1000}],AllTrue[#,PrimeQ]&]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 16 2015 *)

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

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

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 *)

cp's OEIS Frontend

A071696 Greater members of twin prime pairs of form (4*k+1,4*k+3), k>0.

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A071698 Lesser members of twin prime pairs of form (4*k+3, 4*k+5), 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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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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Mathematica

A071696 Greater members of twin prime pairs of form (4k+1,4k+3), k>0.

A071698 Lesser members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

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