A002145 -id:A002145 - cp's OEIS frontend

A080148 Positions of primes of the form 4*k+3 (A002145) among all primes (A000040).

2, 4, 5, 8, 9, 11, 14, 15, 17, 19, 20, 22, 23, 27, 28, 31, 32, 34, 36, 38, 39, 41, 43, 46, 47, 48, 49, 52, 54, 56, 58, 61, 63, 64, 67, 69, 72, 73, 75, 76, 81, 83, 85, 86, 90, 91, 92, 93, 94, 95, 96, 99, 101, 103, 105, 107, 109, 111, 114, 115, 117, 118, 120, 124, 125, 128

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

It appears that a(n) = k such that binomial(prime(k),3) mod 2 = 1. See Maple code. - Gary Detlefs, Dec 06 2011
The above is correct (work mod 4). - Charles R Greathouse IV, Dec 06 2011
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Zak Seidov, Table of n, a(n) for n = 1..10000

Almost complement of A080147 (1 is excluded from both).

Cf. A000040, A002145, A049084.

pos_of_primes_k_mod_n(300,3,4); # Given in A080147.
A080148 := proc(n)
        numtheory[pi](A002145(n)) ;
end proc:
seq(A080148(n),n=1..40) ; # R. J. Mathar, Dec 08 2011

Flatten[Position[Prime[Range[200]],?(IntegerQ[(#-3)/4]&)]] (* _Harvey P. Dale, Jun 06 2011 *)
Select[Range[135], Mod[Prime[#], 4] == 3 &] (* Amiram Eldar, Mar 01 2021 *)

i=0;forprime(p=2,1e3,i++;if(p%4==3,print1(i", "))) \\ Charles R Greathouse IV, Dec 06 2011

a(n) = A049084(A002145(n)). - R. J. Mathar, Oct 06 2008

A385165 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 2+-i modulo p in Gaussian integers.

Original entry on oeis.org

8, 48, 30, 180, 528, 96, 1848, 2208, 1740, 1496, 360, 1560, 2296, 10608, 11448, 5376, 4290, 1932, 11400, 8856, 27888, 16020, 1216, 3300, 3710, 49728, 51528, 14280, 3150, 69168, 7344, 80088, 8568, 48360, 13695, 40136, 6444, 44896, 7980, 146688, 29260, 92880, 48180

Offset: 1

Jianing Song, Jun 20 2025

A002145 are precisely the rational primes in the ring of Gaussian integers.
From the representation of complex numbers as 2 X 2 matrices, a(n) is also the multiplicative order of the matrix [2,-1;1,2] or [2,1;-1,2] modulo p.
a(n) is divisible by ord(5,p): If (2+-i)^n == 1 (mod p), then 5^n == 1 (mod p).
a(n) divides (p+1) * ord(5,p), since we have (2+-i)^(p+1) == 5 (mod p).
If 5 is a quadratic residue modulo p, then ord(5,p) divides (p-1)/2, and so a(n) divides (p^2-1)/2. Conversely, if a(n) divides (p^2-1)/2, then (x+-y*i)^2 == 2+-i (mod p) for some integers x, y, and so (x^2+y^2)^2 == 5 (mod p), which means that 5 is a quadratic residue modulo p.

			The multiplicative order of 2+-i modulo A002145(3) = 11 is a(3) = 30, since (2+-i)^30 == 1 (mod 11), and 30 is the smallest such exponent.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002145, A211241, A385163 (multiplicative order of 1+-i), A385166.

ord(p) = my(d = divisors((p+1)*znorder(Mod(5,p)))); for(i=1, #d, if(Mod([2,-1;1,2],p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i,p)
forprime(p=3, 1e3, if(p%4==3, print1(ord(p), ", ")))

A334426 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^3).

Original entry on oeis.org

1, 0, 4, 1, 1, 5, 8, 0, 7, 2, 8, 2, 3, 4, 4, 4, 5, 8, 0, 3, 3, 8, 3, 6, 0, 5, 6, 9, 9, 2, 5, 6, 1, 5, 6, 6, 9, 3, 7, 6, 0, 7, 1, 3, 5, 1, 1, 3, 4, 9, 3, 5, 4, 1, 7, 3, 9, 4, 9, 8, 8, 6, 6, 6, 1, 7, 8, 5, 4, 1, 3, 5, 5, 8, 5, 6, 1, 3, 5, 0, 3, 5, 3, 5, 6, 0, 4, 7, 4, 5, 5, 4, 6, 7, 1, 0, 8, 7, 4, 3, 1, 5, 3, 6, 3

Offset: 1

Vaclav Kotesovec, Apr 30 2020

			1.041158072823444580338360569925615669376071...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.

Cf. A002145, A243381, A334447, A334451.

A334426 / A334427 = 28*zeta(3)/Pi^3.

A334424 * A334426 = 840*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

A334427 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^3).

Original entry on oeis.org

9, 5, 9, 1, 4, 2, 7, 1, 1, 0, 4, 3, 2, 0, 7, 3, 4, 4, 9, 9, 9, 7, 0, 5, 9, 1, 3, 7, 5, 0, 2, 0, 9, 8, 1, 5, 3, 6, 5, 4, 2, 3, 6, 5, 9, 7, 7, 4, 4, 5, 7, 1, 0, 6, 3, 4, 8, 6, 2, 6, 6, 4, 3, 2, 8, 0, 6, 8, 5, 4, 9, 8, 8, 3, 8, 6, 4, 2, 2, 3, 8, 9, 3, 4, 1, 2, 3, 9, 3, 7, 7, 5, 3, 7, 4, 3, 9, 7, 1, 3, 5, 8, 1, 1, 1, 3

Offset: 0

Vaclav Kotesovec, Apr 30 2020

			0.959142711043207344999705913750209815365423...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 3 3 = 1/A334427).

Cf. A002145, A243379, A334448, A334452.

A334426 / A334427 = 28*zeta(3)/Pi^3.

A334425 * A334427 = 8/(7*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 3, 5, 9, 13, 15, 7, 27, 25, 39, 11, 45, 17, 21, 65, 81, 29, 75, 19, 117, 35, 33, 23, 135, 169, 51, 125, 63, 37, 195, 31, 243, 55, 87, 91, 225, 41, 57, 85, 351, 53, 105, 43, 99, 325, 69, 47, 405, 49, 507, 145, 153, 61, 375, 143, 189, 95, 111, 59, 585, 73, 93, 175, 729, 221, 165, 67, 261, 115, 273, 71, 675, 89, 123

Offset: 1

Antti Karttunen, Nov 02 2021

Permutation of odd numbers. Preserves the prime signature.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000720, A002144, A002145, A348744, A348747 (left inverse).

Cf. also A003961, A332818 for similar maps.

A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Fully multiplicative with a(p) = A348744(A000720(p)), where A348744 is the lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime of the same form.

A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 13, 22, 37, 71, 128, 231, 440, 807, 1519, 2872, 5371, 10204, 19341, 36759, 70179, 134241, 256856, 492936, 947272, 1822615, 3513691, 6781495, 13103816, 25348667, 49092241, 95168205, 184661253, 358636497, 697094872, 1356052491, 2639893495, 5142817901

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A002145, A036378, A095007, A095010, A095012, A095092, A095093.

a(n) = A036378(n) - A095007(n) = A095010(n) + A095012(n) = A095092(n) + A095093(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A107978 Products of two primes of the form 4n+3 (A002145).

Original entry on oeis.org

9, 21, 33, 49, 57, 69, 77, 93, 121, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 361, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, 501, 517, 529, 537, 553, 573, 581, 589, 597, 633, 649, 669, 681, 713, 717, 721, 737

Offset: 1

Jonathan Vos Post, Jun 12 2005

Every odd semiprime must be in one of three disjoint sets: the product of two primes of the form 4n+1 (A121387), the product of two primes of the form 4n+3 (A107978), or the product of a prime of the form 4n+1 and a prime of the form 4n+3 (A080774).

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002145, A080774, A121387.
Union of A131574 and A080109.
Third row of A121388.

p = Select[ Prime@ Range@ 60, Mod[ #, 4] == 3 &]; Take[ Sort@ Flatten@ Table[ p[[i]] p[[j]], {j, 30}, {i, j}], 54] (* or *)
fQ[n_] := Block[{fi = FactorInteger@ n}, Plus @@ Last /@ fi == 2 && Union@ Mod[ First /@ fi, 4] == {3}]; Select[ Range@ 748, fQ@# &] (* Robert G. Wilson v, May 20 2010 *)

{a(n)} = {p*q: p and q both elements of A002145}.

Edited by N. J. A. Sloane, May 20 2010

A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

Original entry on oeis.org

2, 5, 3, 13, 17, 7, 11, 29, 37, 19, 41, 23, 31, 53, 61, 43, 73, 47, 89, 97, 59, 101, 109, 67, 71, 79, 113, 137, 83, 103, 149, 157, 107, 173, 127, 181, 131, 193, 197, 139, 229, 151, 233, 163, 167, 241, 257, 269, 277, 179, 191, 281, 199, 293, 211, 313, 223, 317, 227, 239, 337, 251, 349, 353, 263, 271, 373, 283, 389

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

After 2, for each n >= 1, swap the places of primes A002144(n) and A002145(n) in A000040.

			For n=2, for which A000040(2) = 3, the first prime of the form 4k+3, we select the first prime of the form 4k+1, which is 5, thus a(2) = 5.
For n=3, for which A000040(3) = 5, the first prime of the form 4k+1, we select the first prime of the form 4k+3, which is 3, thus a(3) = 3.
For n=4, for which A000040(4) = 7, the second prime of the form 4k+3, we select the second prime of the form 4k+1, which is 13, thus a(4) = 13.
For n=5, for which A000040(5) = 11, the third prime of the form 4k+3, we select the third prime of the form 4k+1, which is 17, thus a(5) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000040, A002144, A002145, A267097, A267098, A267099, A267100.

Cf. also A108546.

(define (A267101 n) (cond ((= 1 n) 2) ((= 1 (modulo (A000040 n) 4)) (A002145 (A267097 n))) (else (A002144 (A267098 n)))))

a(1) = 2; after which, if prime(n) modulo 4 = 1, a(n) = A002145(A267097(n)), otherwise a(n) = A002144(A267098(n)).
a(n) = A000040(A267100(n)).
a(n) = A267099(A000040(n)).

A334448 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^4).

Original entry on oeis.org

9, 8, 7, 1, 6, 2, 6, 2, 5, 4, 2, 2, 2, 2, 6, 8, 5, 6, 4, 8, 2, 7, 0, 1, 2, 6, 4, 5, 7, 7, 3, 7, 0, 8, 2, 7, 7, 2, 4, 0, 3, 2, 7, 9, 7, 2, 9, 2, 8, 2, 4, 1, 4, 7, 4, 3, 4, 8, 3, 2, 6, 5, 0, 8, 5, 5, 7, 3, 0, 8, 9, 4, 7, 5, 6, 6, 7, 0, 0, 1, 8, 8, 9, 0, 8, 4, 1, 5, 0, 4, 9, 9, 8, 9, 0, 7, 3, 3, 4, 7, 7, 0, 3, 5, 3, 6

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)).

			0.98716262542222685648270126457737082772403279729282414743483...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 3 4 = 1/A334448).

Cf. A002145, A243379, A334427, A334452.

A334447 / A334448 = 1/(PolyGamma(3, 1/4)/(8*Pi^4) - 1).

A334446 * A334448 = 96/Pi^4.

More digits from Vaclav Kotesovec, Jun 27 2020

A082075 First differences of primes of the form 4*k+3 (A002145).

Original entry on oeis.org

4, 4, 8, 4, 8, 12, 4, 12, 8, 4, 8, 4, 20, 4, 20, 4, 8, 12, 12, 4, 12, 12, 8, 12, 12, 4, 12, 12, 12, 8, 12, 24, 4, 20, 16, 12, 8, 12, 4, 36, 12, 8, 4, 20, 4, 12, 8, 4, 8, 4, 20, 24, 16, 8, 16, 12, 8, 12, 12, 12, 4, 12, 24, 8, 28, 8, 12, 4, 8, 36, 24, 12, 4, 12, 20, 4, 20, 4, 20, 4, 8, 28, 20, 4

Offset: 1

Labos Elemer, Apr 07 2003

nonn

			The first and second primes of the form 4*k+3 are 3 and 7, so a(1) = 7-3 = 4.

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

Cf. A002144, A002145, A082073, A082074, A082075, A082076.

k=0; m=4; r=3; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[ep-rp]; ], {n, 1, 1000}]

a(n) = A002145(n+1) - A002145(n).

cp's OEIS Frontend

A080148 Positions of primes of the form 4*k+3 (A002145) among all primes (A000040).

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A385165 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 2+-i modulo p in Gaussian integers.

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A334426 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^3).

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A334427 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^3).

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A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

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A095008 Number of 4k+3 primes (A002145) in range ]2^n,2^(n+1)].

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A107978 Products of two primes of the form 4n+3 (A002145).

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Mathematica

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A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

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