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Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).
(Formerly M3430 N1391)
+20
15
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
REFERENCES
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).
EXAMPLE
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;
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(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).
+20
15
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
COMMENTS
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.
EXAMPLE
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;
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
.................................
EXTENSIONS
Edited. New name, moved the old one to the comment section. - Wolfdieter Lang, Jan 13 2015
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 >= ....
+20
14
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
COMMENTS
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
FORMULA
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/ A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020
EXAMPLE
1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.
MATHEMATICA
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 *)
PROG
(PARI) 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]^2<lim, u=Set(concat(Vec(v), u)); v=List());
);
Set(concat(Vec(v), u));
(Python)
"""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=", ")
AUTHOR
Bernard Altschuler (Altschuler_B(AT)bls.gov), May 30 2000
a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).
+20
14
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
COMMENTS
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
EXAMPLE
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;
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
.................................
MATHEMATICA
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 *)
EXTENSIONS
Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015
Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.
(Formerly M2442 N0970)
+20
13
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
REFERENCES
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).
EXAMPLE
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;
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
.................................
EXTENSIONS
Corrected definition to require p= A002144(n), which defines the order of the terms. - M. F. Hasler, Feb 24 2009
Square of primes of the form 4k+1 ( A002144).
+20
11
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
COMMENTS
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...
REFERENCES
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.
FORMULA
Product_{n>=1} (1 + 1/a(n)) = A243380
Product_{n>=1} (1 - 1/a(n)) = A088539. (End)
EXAMPLE
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
PROG
(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^2" ")) ) }
EXTENSIONS
Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015
Positions of primes of the form 4*k+1 ( A002144) among all primes ( A000040).
+20
11
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
COMMENTS
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021
FORMULA
Numbers k such that prime(k) AND 2 = 0. - Gary Detlefs, Dec 26 2011
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Select[Range[135], Mod[Prime[#], 4] == 1 &] (* Amiram Eldar, Mar 01 2021 *)
CROSSREFS
Almost complement of A080148 (1 is excluded from both).
Decimal expansion of Product_{k>=1} (1 + 1/ A002144(k)^3).
+20
11
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
REFERENCES
B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.
EXAMPLE
1.008761284276077638565924191966917577926199...
First difference set of primes with 4k+1 form: A002144.
+20
10
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
COMMENTS
a(n) is divisible by 4, for all n.
EXAMPLE
first and second 4k+1 primes are 5 and 13, so a(1)=13-5=8;
MATHEMATICA
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 *)
Decimal expansion of Product_{k>=1} (1 - 1/ A002144(k)^3).
+20
10
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
REFERENCES
B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.
EXAMPLE
0.991251116234099844239776364609097744339415...
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