A002144 -id:A002144 - cp's OEIS frontend

A002365 Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).

4, 12, 15, 21, 35, 40, 45, 60, 55, 80, 72, 99, 91, 112, 105, 140, 132, 165, 180, 168, 195, 221, 208, 209, 255, 260, 252, 231, 285, 312, 308, 288, 299, 272, 275, 340, 325, 399, 391, 420, 408, 351, 425, 380, 459, 440, 420, 532, 520, 575, 465, 551, 612, 608, 609

Offset: 1

N. J. A. Sloane

nonn

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
3^2 + 4^2 = 5^2, giving x=3, y=4, p=5 and we have the first terms of A002366, the present sequence and A002144.

A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927 [Annotated scanned copy of selected pages]

Cf. A002144, A002366, A376429.

More terms from Ray Chandler, Jun 23 2004

Revised definition from M. F. Hasler, Feb 24 2009

A002366 Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.

Original entry on oeis.org

3, 5, 8, 20, 12, 9, 28, 11, 48, 39, 65, 20, 60, 15, 88, 51, 85, 52, 19, 95, 28, 60, 105, 120, 32, 69, 115, 160, 68, 25, 75, 175, 180, 225, 252, 189, 228, 40, 120, 29, 145, 280, 168, 261, 220, 279, 341, 165, 231, 48, 368, 240, 35, 105, 200, 315, 300, 385, 52, 260, 259

Offset: 1

N. J. A. Sloane

nonn

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. J. C. Cunningham, Quadratic and Linear Tables, Hodgson, London, 1927 [Annotated scanned copy of selected pages]
Hugo Pfoertner, Plot of log(a(n)) vs log(A002144(n)) using Plot 2.

Cf. A002313, A002330, A002331, A376428.

More terms from Ray Chandler, Jun 23 2004

Corrected definition to require p=A002144(n), which defines the order of the terms. - M. F. Hasler, Feb 24 2009

A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

Original entry on oeis.org

1, 3, 2, 5, 3, 10, 7, 15, 12, 20, 18, 5, 15, 28, 22, 35, 33, 13, 45, 42, 7, 15, 52, 30, 8, 65, 63, 40, 17, 78, 77, 72, 45, 68, 63, 85, 57, 10, 30, 105, 102, 70, 42, 95, 55, 110, 105, 133, 130, 12, 92, 60, 153, 152, 50, 143, 75, 138, 13, 65, 165, 27, 117, 190, 150, 187, 143, 70

Offset: 1

Lekraj Beedassy, May 06 2002

Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values x*y/2.

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
n = 7: a(7) = 7, A002144(7) = 53 and 53^2 = 2809 = A070079(7)^2 + (4*a(7))^2 = 45^2 + (4*7)^2 = 2025 + 784. - _Wolfdieter Lang_, Jan 13 2015

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

Cf. A002144, A002330, A002331, A070079, A080109, A144954, A144960.

a(n) = A002330(n+1)*A002331(n+1)/2. - David Wasserman, May 12 2003

4*a(n) is the even positive integer with A080109(n) = A002144(n)^2 = A070079(n)^2 + (4*a(n))^2 in this unique decomposition (up to order). See A080109 for references. - Wolfdieter Lang, Jan 13 2015

Edited. New name, moved the old one to the comment section. - Wolfdieter Lang, Jan 13 2015

A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....

Original entry on oeis.org

1, 5, 25, 65, 125, 325, 625, 1105, 1625, 3125, 4225, 5525, 8125, 15625, 21125, 27625, 32045, 40625, 71825, 78125, 105625, 138125, 160225, 203125, 274625, 359125, 390625, 528125, 690625, 801125, 1015625, 1185665, 1221025, 1373125, 1795625

Offset: 1

Bernard Altschuler (Altschuler_B(AT)bls.gov), May 30 2000

This sequence is related to Pythagorean triples regarding the number of hypotenuses which are in a particular number of total Pythagorean triples and a particular number of primitive Pythagorean triples.
Least integer "mod 4 prime signature" values that are the hypotenuse of at least one primitive Pythagorean triple. - Ray Chandler, Aug 26 2004
See A097751 for definition of "mod 4 prime signature"; terms of A097752 with all prime factors of form 4*k+1.
Sequence A006339 (Least hypotenuse of n distinct Pythagorean triangles) is a subset of this sequence. - Ruediger Jehn, Jan 13 2022

			1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A097752.

Cf. A002144, A006278, A006339, A071383, A097751, A097752, A097753, A097754, A097755, A097756.

maxTerm = 10^15;(* this limit gives ~ 500 terms *) maxNumberOfExponents = 9;(* this limit has to be increased until the number of reaped terms no longer changes *) bmax = Ceiling[ Log[ maxTerm]/Log[q]]; q = Reap[For[k = 0 ; cnt = 0, cnt <= maxNumberOfExponents, k++, If[PrimeQ[4*k + 1], Sow[4*k + 1]; cnt++]]][[2, 1]]; Clear[b]; b[maxNumberOfExponents + 1] = 0; iter = Sequence @@ Table[{b[k], b[k + 1], bmax[[k]]}, {k, maxNumberOfExponents, 1, -1}]; Reap[ Do[an = Product[q[[k]]^b[k], {k, 1, maxNumberOfExponents}]; If[an <= maxTerm, Print[an]; Sow[an]], Evaluate[iter]]][[2, 1]] // Flatten // Union (* Jean-François Alcover, Jan 18 2013 *)

list(lim)=
{
  my(u=[1], v=List(), w=v, pr, t=1);
  forprime(p=5,,
    if(p%4>1, next);
    t*=p;
    if(t>lim, break);
    listput(w,t)
  );
  for(i=1,#w,
    pr=1;
    for(e=1,logint(lim\=1,w[i]),
      pr*=w[i];
      for(j=1,#u,
        t=pr*u[j];
        if(t>lim, break);
        listput(v,t)
      )
    );
    if(w[i]^2Charles R Greathouse IV, Dec 11 2016

def generate_A054994():
    """generate arbitrarily many elements of the sequence.
    TO_DO is a list of pairs (radius, exponents) where
    "exponents" is a weakly decreasing sequence, and
    radius == prod(prime_4k_plus_1(i)**j for i,j in enumerate(exponents))
    An example entry is (5525, (2, 1, 1)) because 5525 = 5**2 * 13 * 17.
    """
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius #, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(radius,exponents):
    # try to increase each exponent by 1 if possible
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            # can add 1 in position i without violating monotonicity
            yield (radius*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0: # add new exponent 1 at the end:
        yield (radius*prime_4k_plus_1(len(exponents)), exponents+(1,))
from sympy import isprime
primes_congruent_1_mod_4 = [5] # will be filled with 5,13,17,29,37,...
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
for n,radius in enumerate(generate_A054994()):
    if n==34:
        print(radius)
        break # print the first 35 elements
    print(radius, end=", ")
# Günter Rote, Sep 12 2023

Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020

More terms from Henry Bottomley, Mar 14 2001

A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

Original entry on oeis.org

3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675

Offset: 1

Lekraj Beedassy, May 06 2002

Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.

Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005

			The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

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

Cf. A002144, A002330, A002331, A080109, A070151

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003

a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015

More terms from Benoit Cloitre, Jan 13 2003

Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015

A080147 Positions of primes of the form 4*k+1 (A002144) among all primes (A000040).

Original entry on oeis.org

3, 6, 7, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135

Offset: 1

Antti Karttunen, Feb 11 2003

The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

			7 is in the sequence because the 7th prime, 17, is of the form 4k+1.
4 is not in the sequence because the 4th prime, 7, is not of the form 4k+1.

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

Almost complement of A080148 (1 is excluded from both).

Cf. A000040, A002144.

with(numtheory,ithprime); pos_of_primes_k_mod_n(300,1,4);
pos_of_primes_k_mod_n := proc(upto_i,k,n) local i,a; a := []; for i from 1 to upto_i do if(k = (ithprime(i) mod n)) then a := [op(a),i]; fi; od; RETURN(a); end;
with(Bits): for n from 1 to 135 do if (And(ithprime(n),2)=0) then print(n) fi od; # Gary Detlefs, Dec 26 2011

Select[Range[135], Mod[Prime[#], 4] == 1 &] (* Amiram Eldar, Mar 01 2021 *)

k=0;forprime(p=2,1e4,k++;if(p%4==1,print1(k", "))) \\ Charles R Greathouse IV, Dec 27 2011

A002144(n) = A000040(a(n)).

Numbers k such that prime(k) AND 2 = 0. - Gary Detlefs, Dec 26 2011

A080109 Square of primes of the form 4k+1 (A002144).

Original entry on oeis.org

25, 169, 289, 841, 1369, 1681, 2809, 3721, 5329, 7921, 9409, 10201, 11881, 12769, 18769, 22201, 24649, 29929, 32761, 37249, 38809, 52441, 54289, 58081, 66049, 72361, 76729, 78961, 85849, 97969, 100489, 113569, 121801, 124609, 139129

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.
a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three...

			a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.
a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publ. No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.

Cf. A002144, A080175. - Wolfdieter Lang, Jan 13 2015

Cf. A070079, A070151, A088539, A243380.

Select[4 Range[96] + 1, PrimeQ]^2 (* Michael De Vlieger, Dec 27 2016 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^2" ")) ) }

a(n) = A002144(n)^2 = A070079(n)^2 + (4*A070151(n))^2, for n >= 1. - Wolfdieter Lang, Jan 13 2015
From Amiram Eldar, Dec 02 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A243380
Product_{n>=1} (1 - 1/a(n)) = A088539. (End)

Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015

A334424 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 30 2020

			1.008761284276077638565924191966917577926199...

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. A002144, A243380, A334445, A334449.

A334424 / A334425 = 105*zeta(3)/(4*Pi^3).

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

a(17)-a(18) from Jinyuan Wang, Apr 30 2020

More digits from Vaclav Kotesovec, Apr 30 2020 and Jun 27 2020

A082073 First difference set of primes with 4k+1 form: A002144.

Original entry on oeis.org

8, 4, 12, 8, 4, 12, 8, 12, 16, 8, 4, 8, 4, 24, 12, 8, 16, 8, 12, 4, 32, 4, 8, 16, 12, 8, 4, 12, 20, 4, 20, 12, 4, 20, 16, 8, 4, 8, 12, 12, 16, 8, 4, 48, 12, 20, 16, 12, 8, 16, 8, 12, 4, 24, 12, 8, 12, 4, 24, 8, 24, 24, 4, 8, 4, 24, 12, 12, 8, 24, 4, 20, 4, 48, 8, 4, 12, 24, 20, 12, 4, 8, 12

Offset: 1

Labos Elemer, Apr 07 2003

a(n) is divisible by 4, for all n.

			first and second 4k+1 primes are 5 and 13, so a(1)=13-5=8;

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

k=0; m=4; r=1; Do[s=Mod[Prime[n], m]; If[Equal[s, r], rp=ep; k=k+1; ep=Prime[n]; Print[(ep-rp)]; ], {n, 1, 1000}]
Differences[Select[Prime[Range[200]],Mod[#,4]==1&]] (* Harvey P. Dale, Feb 05 2020 *)

p=5;forprime(q=7,1e3,if(q%4==1,print1(q-p", ");p=q)) \\ Charles R Greathouse IV, May 13 2012

a(n) = A002144(n+1) - A002144(n).

A334425 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Apr 30 2020

			0.991251116234099844239776364609097744339415...

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 1 3 = 1/A334425).

Cf. A002144, A088539, A334446, A334450.

A334424 / A334425 = 105*zeta(3)/(4*Pi^3).

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

More digits from Vaclav Kotesovec, Jun 27 2020

cp's OEIS Frontend

A002365 Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).

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A002366 Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.

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A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....

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A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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A080147 Positions of primes of the form 4*k+1 (A002144) among all primes (A000040).

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A080109 Square of primes of the form 4k+1 (A002144).

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