A071695 -id:A071695 - 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

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

A112591 a(n) = prime(n) XOR prime(n + 1).

Original entry on oeis.org

1, 6, 2, 12, 6, 28, 2, 4, 10, 2, 58, 12, 2, 4, 26, 14, 6, 126, 4, 14, 6, 28, 10, 56, 4, 2, 12, 6, 28, 14, 252, 10, 2, 30, 2, 10, 62, 4, 10, 30, 6, 10, 126, 4, 2, 20, 12, 60, 6, 12, 6, 30, 10, 506, 6, 10, 2, 26, 12, 2, 62, 22, 4, 14, 4, 118, 26, 10, 6, 60, 6, 8, 26, 14, 4, 250, 8, 28, 8

Offset: 1

Sandeep Chellappen (sandeep.chellappen(AT)gmail.com), Dec 18 2005

When represented in binary, this sequence represents the uncommon bits in two consecutive prime numbers.

a(n) = 2 for indices of the lesser twin primes of the form 4k + 1, A071695. - Michel Marcus, Apr 17 2020

			a(2) = 6 ; since prime(2) = 3, which is 11 in binary, prime(3) = 5, which is 101 in binary; and 011 XOR 101 = 110, which is 6 in decimal.

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

Cf. A003987, A071695, A375235.

A112591 := proc(n) local ndual,n2dual,nxor,i ; ndual := convert(ithprime(n),base,2) ; n2dual := convert(ithprime(n+1),base,2) ; nxor := [] ; i := 1 ; while i <= nops(ndual) do nxor := [op(nxor), abs(op(i,ndual)-op(i,n2dual)) ] ; i := i+1 ; od ; while i <= nops(n2dual) do nxor := [op(nxor), op(i,n2dual) ] ; i := i+1 ; od ; add( op(i,nxor)*2^(i-1),i=1..nops(nxor)) ; end: for n from 1 to 80 do printf("%d,",A112591(n)) ; od ; # R. J. Mathar, Mar 07 2007
with(Bits):seq(Xor(ithprime(n),ithprime(n+1)),n=1..50) # Gary Detlefs, Aug 03 2013

BitXor@@#&/@Partition[Prime[Range[80]], 2, 1] (* Harvey P. Dale, May 04 2018 *)

a(n) = bitxor(prime(n),prime(n+1)); \\ Joerg Arndt, Aug 04 2013

val prime: LazyList[Int] = 2 #:: LazyList.from(3).filter(i => prime.takeWhile { j => j * j <= i }.forall { k => i % k != 0 })
(0 to 127).map(n => prime(n) ^ prime(n + 1)) // Alonso del Arte, Apr 18 2020

More terms and better name from Christopher M. Herron (cmh285(AT)psu.edu), Apr 25 2006

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.

A205172 Primes p == 5 (mod 8) such that p + 2 is also prime.

Original entry on oeis.org

5, 29, 101, 149, 197, 269, 461, 821, 1061, 1229, 1277, 1301, 1877, 1949, 1997, 2141, 2237, 2309, 2381, 2549, 2789, 3389, 3461, 3557, 3581, 3821, 3917, 4157, 4229, 4421, 4517, 4637, 5021, 5477, 5501, 5741, 6197, 6269, 6701, 6869, 7349, 7589, 7757, 7877

Offset: 1

Robert G. Wilson v, Jan 22 2012

The lesser of twin primes == 5 (mod 8).

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

Cf. A071695, A205171.

select(t -> isprime(t) and isprime(t+2), [seq(i,i=5..10000,8)]);# Robert Israel, Nov 25 2019

Select[ Prime@ Range@ 1000, Mod[#, 8] == 5 && PrimeQ[# + 2] &]

forprime(p=1, 7900, if(Mod(p, 8)==5 && ispseudoprime(p+2), print1(p, ", "))) \\ Felix Fröhlich, Nov 25 2019

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.

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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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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A112591 a(n) = prime(n) XOR prime(n + 1).

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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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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.

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.

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.