A002144 -id:A002144 - cp's OEIS frontend

A096030 Values (y-x-1)/2, where x^2+y^2=p,(xA002144.

0, 0, 1, 1, 2, 0, 2, 0, 2, 1, 2, 4, 3, 0, 3, 1, 2, 5, 0, 2, 6, 6, 2, 5, 7, 1, 2, 5, 7, 0, 1, 3, 6, 4, 5, 3, 6, 9, 8, 0, 2, 6, 8, 4, 8, 4, 5, 2, 3, 11, 7, 9, 0, 1, 10, 4, 9, 5, 12, 10, 3, 12, 8, 0, 6, 2, 7, 11, 5, 8, 2, 12, 11, 4, 1, 2, 9, 7, 13, 12, 6, 0, 9, 14, 13, 10, 8, 15, 6, 1, 2, 3, 12, 14, 9, 0, 2

Offset: 1

Lekraj Beedassy, Jun 16 2004

nonn

Cf. A005098, A096029.

a(n)=(A079887(n) - 1)/2.

More terms from Ray Chandler, Jun 26 2004

A173330 First of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

Original entry on oeis.org

1, 10, 1, 5, 1, 5, 46, 5, 70, 5, 9, 1, 106, 106, 126, 142, 146, 13, 9, 186, 1, 214, 13, 226, 1, 13, 9, 5, 17, 13, 306, 9, 5, 17, 366, 17, 378, 1, 406, 406, 17, 442, 21, 442, 5, 510, 21, 538, 13, 1, 570, 5, 17, 598, 25, 13, 25, 650, 1, 5, 694, 706, 9, 742, 25, 17, 786, 5, 25

Offset: 1

Reinhard Zumkeller, Feb 16 2010

nonn

A002972(n) = MIN(a(n), A002144(n) - a(n)).

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = 26! / (2*(13!)^2) mod 53 = 403291461126605635584000000/77551576087265280000 mod 53 = 5200300 mod 53 = 46,
A002972(7) = MIN(46, 53 - 46) = 7;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = 30! / (2*(15!)^2) mod 61 = 265252859812191058636308480000000/3420024505448398848000000 mod 61 = 77558760 mod 61 = 5,
A002972(8) = MIN(5, 61 - 5) = 5.

H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.

Cf. A173331, A001700, A010050, A000142, A005098.

a(n) = (2k)! / 2(k!)^2 mod p, where p = 4*k+1 = A002144(n).

A231754 Products of distinct primes congruent to 1 modulo 4 (A002144).

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 293, 305, 313, 317, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 433, 445, 449

Offset: 1

Michel Marcus, Nov 13 2013

Contains A002144 as a subsequence, and is a subsequence of A016813 and of A005117.

Also, these numbers satisfy A231589(n) = floor(n*(n-1)/4) (A011848).

			65 = 5*13 is in the sequence since both 5 and 13 are congruent to 1 modulo 4.

R. J. Mathar, Table of n, a(n) for n = 1..4999
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
Shailesh A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag., Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Intersection of A005117 and A004613.
Cf. A002144, A011848, A016813, A167181, A231589.
Cf. A088539, A175647.

isA231754 := proc(n)
    local d;
    for d in ifactors(n)[2] do
        if op(2,d) > 1 then
            return false;
        elif modp(op(1,d),4) <> 1 then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 500 do
    if isA231754(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 16 2016

Select[Range[500], # == 1 || AllTrue[FactorInteger[#], Last[#1] == 1 && Mod[First[#1], 4] == 1 &] &] (* Amiram Eldar, Mar 08 2024 *)

isok(n) = if (! issquarefree(n), return (0)); if (n > 1, f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 1, return (0)))); 1

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A088539 * sqrt(A175647) / Pi = 0.3097281805... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

A095007 Number of 4k+1 primes (A002144) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 10, 21, 38, 66, 127, 233, 432, 805, 1511, 2837, 5378, 10186, 19294, 36827, 70157, 133975, 256852, 492882, 946848, 1823129, 3513599, 6780412, 13103462, 25348870, 49090415, 95167380, 184662052, 358630671, 697097364, 1356051342, 2639870329, 5142823877

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

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. A002144, A036378, A095008, A095009, A095011.

a(n) = A036378(n) - A095008(n) = A095009(n) + A095011(n).

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

A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).

Original entry on oeis.org

6, 30, 60, 210, 210, 180, 630, 330, 1320, 1560, 2340, 990, 2730, 840, 4620, 3570, 5610, 4290, 1710, 7980, 2730, 6630, 10920, 12540, 4080, 8970, 14490, 18480, 9690, 3900, 11550, 25200, 26910, 30600, 34650, 32130, 37050, 7980, 23460, 6090, 29580, 49140, 35700

Offset: 1

M. F. Hasler, Feb 24 2009

nonn

Pythagorean primes, i.e., primes of the form p = 4k+1 = A002144(n), have exactly one representation as sum of two squares: A002144(n) = x^2+y^2 = A002330(n+1)^2+A002331(n+1)^2. The corresponding (primitive) integer-sided right triangle with sides { 2xy, |x^2-y^2| } = { A002365(n), A002366(n) } has area xy|x^2-y^2| = a(n). For n>1 this is a(n) = 30*A068386(n).

			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, A002365, A002366, A144954.

Reap[For[p = 2, p < 500, p = NextPrime[p], If[Mod[p, 4] == 1, area = x*y/2 /. ToRules[Reduce[0 < x <= y && p^2 == x^2 + y^2, {x, y}, Integers]]; Sow[area]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2015 *)

forprime(p=1,499, p%4==1 | next; t=[p,lift(-sqrt(Mod(-1,p)))]; while(t[1]^2>p,t=[t[2],t[1]%t[2]]); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))

{Q=Qfb(1,0,1);forprime(p=1,499,p%4==1|next;t=qfbsolve(Q,p); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))} \\ David Broadhurst

a(n) = A002365(n)*A002366(n)/2.

a(n) = x*y*(x^2-y^2), where x = A002330(n+1), y = A002331(n+1).

A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)... A002144(n).

Original entry on oeis.org

2, 8, 47, 2163, 18543, 241727, 3101272, 842894268, 8245041748, 521781374353, 101476250977928, 671795954794788, 32126984574675193, 425090834074746637, 309609468228403885693, 25836182225971546313682, 38544366727563360743217, 217758730168965028986551783, 25789605237863389220212237968, 309600287787935978580674202007

Offset: 1

Michel Lagneau, Jan 28 2011

nonn

			a(1) = 2 because A002144(1) | 2^2+1 = 5 ;
a(2)=8 because A002144(1) * A002144(2) | 8^2+1 = 5*13 ;
a(6) = 241727 because A002144(1) * A002144(2)*...* A002144(6) | 241727^2+1
  = 2 * 5 * 13 * 17 * 29 * 37 * 41 * 601.

Cf. A002144 (Pythagorean primes: primes of form 4n+1) A002731.

with(numtheory):nn:=1000:T:=array(1..1000):k:=1:for x from 1 to nn do: p:=4*x+1:if
  type(p, prime)=true then T[k]:=p:k:=k+1:else fi:od:pr:=1:for n from 1 to k do:
  pp:=pr*T[n] :ind:=0:for q from 1 to pp while (ind=0) do: z:=q^2+1:if irem(z,pp)=0
  and ind = 0 then ind: = 1:pr:=pp:print( q):else fi:od:od:
# Alternative
PP:= select(isprime, [seq(i,i=5..200,4)]):
f:= n -> min(map(t -> rhs(op(t)),[msolve(x^2+1, convert(PP[1..n],`*`))])):
map(f, [$1..20]); # Robert Israel, Feb 01 2019

More terms from Robert Israel, Feb 01 2019

A247384 Find the first (maximal) string of consecutive primes of length exactly n which alternate between 4k+1 and 4k+3 or 4k+3 and 4k+1 as in A002144(4n+1) and A002145(4n+3). The first element is a(n).

Original entry on oeis.org

97, 11, 3, 23, 47, 167, 131, 2011, 233, 23633, 34499, 1013, 9341, 90659, 521, 51749, 505049, 1391087, 2264839, 2556713, 17123893, 2569529, 15090641, 18246451, 6160043, 1557431471, 43679609, 198572029, 701575297, 5552898499, 6639843979, 61233611783, 9005520203

Offset: 1

J. M. Bergot, Sep 15 2014

nonn

			a(4)=23 because 23,29,31,37 alternate 4*n+3,4*n+1,4*n+3,4*n+1 for exactly four primes and 23 is the least prime for a string of exactly four.

Jens Kruse Andersen and Giovanni Resta, Table of n, a(n) for n = 1..45 (first 38 terms from Jens Kruse Andersen)
Jens Kruse Andersen, Consecutive Congruent Primes

Cf. A002144, A002145, A098058, A098059, A289118, A289237.

Primes:= select(isprime,[seq(2*i+1,i=1..10^7)]):
Pm4:= map(`modp`,[seq((-1)^j*Primes[j],j=1..nops(Primes))],4):
Starts:= [1,op(select(t -> Pm4[t-1]<> Pm4[t], [$2..nops(Pm4)]))]:
Lengths:= [seq(Starts[i+1]-Starts[i],i=1..nops(Starts)-1)]:
for i from 1 to max(Lengths) do A[i]:= ListTools:-Search(i,Lengths) od:
R:=[seq(A[i],i=1..max(Lengths))]:
seq(`if`(a=0,0,Primes[Starts[a]]),a=R); # Robert Israel, Sep 15 2014

i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 4] != 0 || Mod[ Prime[i + 1] - Prime[i], 4] != 0, i++]; T = {Prime[i]}; Do[j = 2; While[! (Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] != 0 && (Mod[Prime[j] - Prime[j - 1], 4] == 0 || j == 2) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 4] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 13}]; T (* Jonathan Sondow, Jun 28 2017 *)

v=vector(100);v[1]=7;cur=1;p=3;forprime(q=5, 1e10, if((q-p)%4==0,if(!v[cur],v[cur]=back(p,cur);print("a("cur") = "v[cur]));cur=1,cur++);p=q) \\ Charles R Greathouse IV, Sep 15 2014

a(n) = A289118(n) if and only if n > 1 and A289118(n) < A289118(n+1). - Jonathan Sondow, Jun 27 2017

More terms from Jens Kruse Andersen, Oct 01 2014

Definition clarified by Jonathan Sondow, Jun 25 2017

A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.

Original entry on oeis.org

7, 65, 161, 41, 1081, 369, 1241, 671, 721, 3471, 959, 9401, 4681, 1695, 3281, 7599, 10199, 24521, 3439, 18335, 37241, 45241, 24465, 29281, 64001, 18561, 31855, 27761, 76601, 7825

Offset: 1

Wolfdieter Lang, Jan 14 2015

The corresponding even legs are given in 4*A253803.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253804(n) (odd) and A253805(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^2 = A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Note that the Pythagorean triangles are not always primitive. E.g., n = 2: (65, 4*39, 13^2) = 13*(5, 4*3, 13). For each prime congruent 1 (mod 4) (A002144) there is one and only one such non-primitive triangle with hypotenuse p^2 (just scale the unique primitive triangle with hypotenuse p with the factor p). Therefore, one of the two existing Pythagorean triangles with hypotenuse from A080109 is primitive and the other is imprimitive.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4  =  a(7)^2 + (4*A253803(7))^2 = 1241^2 + (4*630)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253305, A253803, A253804.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253803(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253803(n))^2), n >= 1.

A253803 a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).

Original entry on oeis.org

6, 39, 60, 210, 210, 410, 630, 915, 1320, 1780, 2340, 990, 2730, 3164, 4620, 5215, 5610, 4290, 8145, 8106, 2730, 6630, 12116, 12540, 4080, 17485, 17451, 18480, 9690, 24414

Offset: 1

Wolfdieter Lang, Jan 14 2015

See A253802 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253802(7)^2 + (4*a(7))^2 = 1241^2 + (4*630)^2.
The other Pythagorean triangle with hypotenuse
53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A080109, A253802, A253804, A253805.

a(n) = sqrt(A080109(n)^2 - A253802(n)^2)/4, n >= 1.

A253804 a(n) gives the odd leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the larger of the two possible odd legs.

Original entry on oeis.org

15, 119, 255, 609, 1295, 1519, 2385, 3479, 4015, 4879, 6305, 9999, 9919, 12319, 14385, 16999, 13345, 28545, 32039, 19199, 38415, 50609, 32239, 50369, 65535, 62839, 50279, 64911, 83505, 96719

Offset: 1

Wolfdieter Lang, Jan 16 2015

The corresponding even legs are given in 4*A253805.
The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253802(n) (odd) and A253803(n) (even).
Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^= A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
Concerning the primitivity question of these triangles see a comment on A253802.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = a(7)^2 + (4*A253805(7))^2 = 2385^2 + (4*371)^2.
The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4*A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.

L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253303, A253802, A253805.

A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253805(n))^2,
n >= 1, that is,
a(n) = sqrt(A080175(n) - (4*A253805(n))^2), n >= 1.

cp's OEIS Frontend

A096030 Values (y-x-1)/2, where x^2+y^2=p,(xA002144.

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A173330 First of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

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A231754 Products of distinct primes congruent to 1 modulo 4 (A002144).

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A095007 Number of 4k+1 primes (A002144) in range ]2^n,2^(n+1)].

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A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).

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A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)*...* A002144(n).

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A247384 Find the first (maximal) string of consecutive primes of length exactly n which alternate between 4*k+1 and 4*k+3 or 4*k+3 and 4*k+1 as in A002144(4*n+1) and A002145(4*n+3). The first element is a(n).

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A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.

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A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)... A002144(n).

A247384 Find the first (maximal) string of consecutive primes of length exactly n which alternate between 4k+1 and 4k+3 or 4k+3 and 4k+1 as in A002144(4n+1) and A002145(4n+3). The first element is a(n).