A071696 -id:A071696 - 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

A072010 In prime factorization of n replace all primes of form k4+1 with k4+3 and primes of form k4+3 with k4+1.

Original entry on oeis.org

1, 2, 1, 4, 7, 2, 5, 8, 1, 14, 9, 4, 15, 10, 7, 16, 19, 2, 17, 28, 5, 18, 21, 8, 49, 30, 1, 20, 31, 14, 29, 32, 9, 38, 35, 4, 39, 34, 15, 56, 43, 10, 41, 36, 7, 42, 45, 16, 25, 98, 19, 60, 55, 2, 63, 40, 17, 62, 57, 28, 63, 58, 5, 64, 105, 18, 65, 76, 21, 70, 69

Offset: 1

Reinhard Zumkeller, Jun 05 2002

a(3^n) = 1; a(2^n) = 2^n;

a(n)>2 is prime iff n=m*3^i (i>=0), a(n)=a(m) and (m,a(m)) or (a(m),m) is a twin prime pair of form ((4*k+1),(4*k+3)), a(m)*m=A071697(j)=A071695(j)*A071696(j) for some j.

			a(26928) = a(2^4*3^2*11*17) = a(2)^4 * a(3)^2 * a(11) * a(17)
= 2^4 * 1^2 * 9 * 19 = 2736.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002144, A002145, A072012(n) = a(a(n)).
For a(n) = n see A072011.
Cf. A072014, A072015.
Cf. A027746.

a072010 1 = 1
a072010 n = product $ map f $ a027746_row n where
   f 2 = 2
   f p = p + 2 * (2 - p `mod` 4)
-- Reinhard Zumkeller, Apr 09 2012

a[1] = 1; a[p_?PrimeQ] = p + 2*(2 - Mod[p, 4]); a[n_] := Times @@ (a[#[[1]]]^#[[2]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 04 2012 *)

Multiplicative with a(p) = p + 2*(2 - p mod 4), p prime.

A071697 Product of twin primes of form (4k+1,4k+3), k>0.

Original entry on oeis.org

35, 323, 899, 1763, 10403, 19043, 22499, 39203, 72899, 79523, 213443, 272483, 324899, 381923, 412163, 656099, 675683, 736163, 777923, 1102499, 1127843, 1512899, 1633283, 1664099, 1695203, 2196323, 2883203, 2965283, 3526883, 3802499, 3992003, 4334723, 4536899

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Cf. A037074, A071700.

Cf. A071695, A071696.

[16*n^2+16*n+3: n in [1..700]| IsPrime(4*n+1) and IsPrime(4*n+3)]; // Vincenzo Librandi, Feb 24 2015

lista(nn) = {forprime(p=3, nn, if ((((p-1) % 4) == 0) && isprime(p+2), print1(p*(p+2), ", ")););} \\ Michel Marcus, Feb 24 2015

a(n) = A071695(n)*A071696(n).

More terms from Michel Marcus, Feb 24 2015

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

A072028 Swap twin prime pairs of form (4k+1,4k+3) in prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 11, 12, 13, 10, 21, 16, 19, 18, 17, 28, 15, 22, 23, 24, 49, 26, 27, 20, 31, 42, 29, 32, 33, 38, 35, 36, 37, 34, 39, 56, 43, 30, 41, 44, 63, 46, 47, 48, 25, 98, 57, 52, 53, 54, 77, 40, 51, 62, 59, 84, 61, 58, 45, 64, 91, 66

Offset: 1

Reinhard Zumkeller, Jun 07 2002

			a(65) = a(5*13) = a(5)*a(13) = a(4*1+1)*a(13) = (4*1+3)*13 = 7*13 = 91.

Cf. A061898, A064505, A071695, A071696, A072026, A072027, A072029.

f[p_, e_] := If[p < 5, p, If[(m = Mod[p, 4]) == 1 && PrimeQ[p + 2], p + 2, If[m == 3 && PrimeQ[p - 2], p - 2, p]]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 26 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; if(p < 5, p, if(p%4 == 1 && isprime(p+2), p+2, if(p%4 == 3 && isprime(p-2), p-2, p)))^f[i,2]);} \\ Amiram Eldar, Feb 26 2024

Multiplicative with a(p) = (if p mod 4 = 1 and p+2 is prime then p+2 else (if p mod 4 = 3 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))) = 1.0627249749498993391... . - Amiram Eldar, Feb 26 2024

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)

A072011 Numbers k such that A072010(k) = k.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 35, 64, 70, 128, 140, 256, 280, 323, 512, 560, 646, 899, 1024, 1120, 1225, 1292, 1763, 1798, 2048, 2240, 2450, 2584, 3526, 3596, 4096, 4480, 4900, 5168, 7052, 7192, 8192, 8960, 9800, 10336, 10403, 11305

Offset: 1

Reinhard Zumkeller, Jun 05 2002

nonn

If A072010(n) = n then also A072010(n*3^i) = n and A072010(n*2^j) = n*2^j.

For m=(4*k+1)*(4*k+3), product of twin prime pairs: A072010(m) = m, as well as for values m in the free monoid generated by the range of A071697.

			395675 is a term as f(395675) = f(323*1225) = f((17*19)*(5*7)^2) = f(17*19)*(f(5*7))^2 = f(17)*f(19)*(f(5)*f(7))^2 = 19*17*(7*5)^2 = 323*1225 = 395675 for f = A072010.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A002144, A002145, A071695, A071696, A071697, A072010.

A071701 Number of twin prime pairs <= n of form (4k+1,4k+3), k>0.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 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, 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.

			a(30)=3, since (29,31) is included along with (5,7) and (17,19).

a(n) = A071538(n) - A071702(n), cf. A071695, A071696.

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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A072010 In prime factorization of n replace all primes of form k*4+1 with k*4+3 and primes of form k*4+3 with k*4+1.

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A071697 Product of twin primes of form (4*k+1,4*k+3), 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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A072028 Swap twin prime pairs of form (4*k+1,4*k+3) in prime factorization of n.

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

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A072011 Numbers k such that A072010(k) = k.

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A071701 Number of twin prime pairs <= n of form (4*k+1,4*k+3), k>0.

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

A072010 In prime factorization of n replace all primes of form k4+1 with k4+3 and primes of form k4+3 with k4+1.

A071697 Product of twin primes of form (4k+1,4k+3), k>0.

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

A072028 Swap twin prime pairs of form (4k+1,4k+3) in prime factorization of n.

A071701 Number of twin prime pairs <= n of form (4k+1,4k+3), k>0.