A002144 -id:A002144 - cp's OEIS frontend

A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

5, 30, 34, 145, 111, 180, 371, 330, 876, 1560, 1746, 505, 1635, 840, 3014, 3570, 5181, 2249, 1710, 7980, 1379, 3435, 10920, 7230, 2056, 8970, 14490, 11240, 4981, 3900

Offset: 1

Wolfdieter Lang, Jan 16 2015

See A253804 for comments and the Dickson reference.

			n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253804(7)^2 + (4*a(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, A080109, A253804, A253802, A253803.

a(n) = sqrt(A080109(n)^2 - A253804(n)^2)/4, n >= 1.

A267859 The p-defect p - N(p) of the elliptic curve y^2 = x^3 + x for primes p congruent to 1 modulo 4 (A002144).

Original entry on oeis.org

2, -6, 2, 10, 2, 10, -14, 10, -6, 10, 18, 2, -6, -14, -22, -14, -22, 26, 18, -14, 2, -30, 26, -30, 2, 26, 18, 10, 34, 26, -22, 18, 10, 34, -14, 34, -38, 2, -6, -30, 34, -14, 42, -38, 10, -22, 42, -38, 26, 2, -46, 10, 34, -38, 50, 26, 50, -46, 2, 10, -30, -54, 18, -38, 50, 34, -22, 10, 50, -54

Offset: 1

Wolfdieter Lang, Feb 06 2016

sign

See A002172 for a differently signed sequence.
The number N(p) of solutions modulo a prime p of the elliptic curve y^2 = x^3 + x (of discriminant -4) is given for all p in A095978.
The p-defect a_p = p - N(p) for prime 2 and primes congruent to 3 modulo 4 vanishes.
A002144(n) - (a(n)/2)^2  = (2*A002973(n))^2, n >= 1. See the formula for A095978 for primes 1 (mod 4).
This sequence gives also the non-vanishing p-defects of the elliptic curve y^2 = x^3 - 4*x. See a comment on A138515 with the Martin and Ono link for the modularity series for these two elliptic curves. - Wolfdieter Lang, May 26 2016

			n = 2: p = A002144(2) = 13 = A000040(6), m = 6, a(2) = 13 - A095978(6) = 13 - 19  = -6.
n = 2:  -6 = A138515((A002144(2) - 1)/4) =
A138515(3) = -6. - _Wolfdieter Lang_, May 26 2016

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 398. In the 4th ed., 2014, p. 371.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A002144, A002172, A095978, A138515.

terms = 100; A002144 = Select[Range[5, 20*terms, 4], PrimeQ]; A095978[n_] := Module[{p, xy, x}, p = Prime[n]; If[n==1 || Mod[p, 4]==3, Return[p]]; xy = {Re[#], Im[#]}& @ FactorInteger[p, GaussianIntegers -> True][[2, 1]]; x = SelectFirst[xy, OddQ]; If[Mod[x, 4]==1, p - 2*x, p + 2*x]]; a[n_] := (p = A002144[[n]]; m = PrimePi[p]; p - A095978[m]); Array[a, terms] (* Jean-François Alcover, Feb 26 2016, after Robert Israel (A095978) *)

a(n) = A002144(n) - A095978(m) with A002144(n) = A000040(m), n >= 1.

a(n) = A138515((A002144(n) - 1)/4), n >= 1. - Wolfdieter Lang, May 26 2016

A340388 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)13^(p_2 - 1)17^(p_3 - 1)...*A002144(k)^(p_k - 1).

Original entry on oeis.org

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 32045, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125, 37252902984619140625

Offset: 1

Jianing Song, Apr 24 2021

Analog of A037019: this is an easy way to produce a number k such that A002654(k) = n, or equivalently, a number k whose prime factors are all congruent to 1 modulo 4 and with exactly n divisors.

			12 = 3 * 2 * 2, so a(12) = 5^(3-1) * 13^(2-1) * 17^(2-1) = 5525.
15 = 5 * 3, so a(15) = 5^(5-1) * 13^(3-1) = 105625.

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A002144, A002654, A018782, A037019, A340624.

a(n) = my(f=factor(n), w=omega(n), p=1, product=1); forstep(i=w, 1, -1, for(j=1, f[i,2], p=nextprime(p+1); while(!(p%4==1), p=nextprime(p+1)); product *= p^(f[i,1]-1))); product

By definition a(n) >= A018782(n) for all n. Note that a(16) = 32045 is strictly larger than A018782(16) = 27625. The "exceptional" numbers k such that a(k) > A018782(k) are listed in A340624.

If n = p for prime p or n = pq for primes p >= q, then a(n) = A018782(n).

A376428 Numbers k that occur as shorter legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

Original entry on oeis.org

3, 5, 8, 9, 11, 12, 15, 19, 20, 25, 28, 29, 32, 35, 39, 40, 45, 48, 49, 51, 52, 59, 60, 61, 65, 68, 69, 71, 72, 75, 79, 80, 85, 88, 95, 101, 105, 108, 112, 115, 120, 121, 129, 131, 132, 139, 140, 141, 145, 148, 159, 160, 165, 168, 169, 171, 175, 180, 181, 188, 189

Offset: 1

Hugo Pfoertner, Sep 22 2024

nonn

Distinct sorted terms of A002366.

Cf. A002144, A002365, A376429, A376430.

Subsequence of A020884.

A376429 Numbers k that occur as longer legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

Original entry on oeis.org

4, 12, 15, 21, 35, 40, 45, 55, 60, 72, 80, 91, 99, 105, 112, 132, 140, 165, 168, 180, 195, 208, 209, 221, 231, 252, 255, 260, 272, 275, 285, 288, 299, 308, 312, 325, 340, 351, 380, 391, 399, 408, 420, 425, 440, 459, 465, 520, 532, 551, 552, 572, 575, 589, 595, 600

Offset: 1

Hugo Pfoertner, Sep 22 2024

nonn

Distinct sorted terms of A002365.

Cf. A002144, A002365, A376428, A376430.

Subsequence of A020883.

A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).

Original entry on oeis.org

125, 2197, 4913, 24389, 50653, 68921, 148877, 226981, 389017, 704969, 912673, 1030301, 1295029, 1442897, 2571353, 3307949, 3869893, 5177717, 5929741, 7189057, 7645373, 12008989, 12649337, 13997521, 16974593, 19465109, 21253933

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the sum of two squares in exactly two ways (Fermat). See the Dickson reference, (B) on p. 277. - Wolfdieter Lang, Jan 15 2015
a(n) is the hypotenuse of three and only three right triangles with integral arms.
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... .
  See the Dickson reference, (A) on p. 227.

			a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
a(2) = 9^2 + 46^2  = 39^2 + 26^2, and these are the only decompositions. - _Wolfdieter Lang_, Jan 15 2015

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

Cf. A002144, A080109, A080175, A334425.

Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* Amiram Eldar, Dec 02 2022 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^3" ")) ) }

a(n) = A002144(n)^3, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334425. - Amiram Eldar, Dec 02 2022

Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 15 2015

A102261 a(n) = A002144(n) - A002145(n).

Original entry on oeis.org

2, 6, 6, 10, 14, 10, 10, 14, 14, 22, 26, 22, 26, 10, 30, 22, 26, 34, 30, 30, 30, 50, 42, 42, 46, 46, 50, 42, 42, 50, 46, 54, 42, 42, 42, 42, 38, 34, 30, 38, 14, 18, 18, 18, 46, 54, 62, 70, 78, 78, 90, 78, 66, 54, 70, 66, 62, 66, 58, 70, 66, 86, 98, 78, 78, 54, 70, 70, 78, 78

Offset: 1

Paul Curtz, Sep 06 2008

a(n) = A108546(2*n+1) - A108546(2*n).

Ivan Neretin, Table of n, a(n) for n = 1..30000

A002144 := proc(n) option remember ; if n = 1 then RETURN(5) ; fi; for a from procname(n-1)+2 do if isprime(a) and (a mod 4 = 1 ) then RETURN(a) ; fi; od: end; A002145 := proc(n) option remember ; if n = 1 then RETURN(3) ; fi; for a from procname(n-1)+2 do if isprime(a) and (a mod 4 = 3 ) then RETURN(a) ; fi; od: end; A102261 := proc(n) A002144(n)-A002145(n) ; end: seq(A102261(n),n=1..120) ; # R. J. Mathar, Feb 07 2009

nmax = 70; a1 = Select[Range[1, Prime[3*nmax], 4], PrimeQ]; a3 = Select[Range[3, Prime[3*nmax], 4], PrimeQ]; a[n_] := a1[[n]] - a3[[n]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 17 2013 *)

Edited by N. J. A. Sloane, Sep 06 2008

More terms from R. J. Mathar, Feb 07 2009

A113601 Intersection of A002144 and A005098.

Original entry on oeis.org

13, 37, 73, 97, 193, 277, 373, 409, 433, 577, 673, 709, 853, 997, 1033, 1093, 1129, 1297, 1429, 1453, 1549, 1597, 1753, 1777, 2017, 2029, 2293, 2437, 2677, 2713, 2953, 3037, 3049, 3109, 3229, 3457, 3469, 3637, 3769, 3793, 3853, 4057, 4273, 4297, 4729

Offset: 1

Zak Seidov, May 27 2007

nonn

Primes p of the form 4m+1 such that q=4p+1 is also prime. Corresponding values of m are: 3,9,18,24,48,69,93,102,108,144,168,177,213,249,258,273,282,324,357,363,387,399,438,444,504,507,573,609,669,678,738,759,762,777,807,864,867,909 - all multiples of 3. And corresponding values of q are: 53,149,293,389,773,1109,1493,1637,1733,2309,2693,2837,3413,3989,4133,4373,4517,5189,5717,5813,6197,6389,7013,7109,8069,8117,9173,9749,10709,10853,11813.

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

Cf. A002144, A005098.

Select[Prime[Range[700]],IntegerQ[(#-1)/4]&&PrimeQ[4#+1]&] (* Harvey P. Dale, May 15 2022 *)

A145871 Smallest k such that k^2+1 is divisible by A002144(n)^7.

Original entry on oeis.org

32318, 6826318, 96940388, 7986582530, 24900904028, 92615568742, 416081467190, 988322434636, 3219884218827, 4867146503697, 26457926739667, 47023298541694, 26661771973542, 90980209992989, 257680081342861, 283410689912607

Offset: 1

Klaus Brockhaus, Oct 22 2008

nonn

			a(2) = 6826318 since A002144(2) = 13, 6826318^2+1 = 46598617437125 = 5^3*13^7*13*457 and for no k < 6826318 does 13^7 divide k^2+1. a(4) = 7986582530 since A002144(4) = 29, 7986582530^2+1 = 63785500508501200901 = 29^7*197*409*45893 and for no k < 7986582530 does 29^7 divide k^2+1.

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298, A145299, A145872, A145873.

{e=7; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}

More terms from Klaus Brockhaus, Nov 12 2008

A145872 Smallest k such that k^2+1 is divisible by A002144(n)^8.

Original entry on oeis.org

110443, 6826318, 3379649772, 61012922706, 1019349744435, 287369842623, 11331029931180, 71294762793847, 239822883201307, 923990886302412, 2369608176604944, 3156215819652023, 521749964271465, 2026364722410364

Offset: 1

Klaus Brockhaus, Oct 22 2008

nonn

			a(1) = 110443 since A002144(1) = 5, 110443^2+1 = 12197656250 = 2*5^8*13*1201 and for no k < 110443 does 5^8 divide k^2+1. a(3) = 3379649772 since A002144(3) = 17, 3379649772^2+1 = 11422032581379651985 = 5*13*17^8*97*259697 and for no k < 3379649772 does 17^8 divide k^2+1.

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298, A145299, A145871, A145873.

{e=8; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}

More terms from Klaus Brockhaus, Nov 12 2008

cp's OEIS Frontend

A253805 a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.

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A267859 The p-defect p - N(p) of the elliptic curve y^2 = x^3 + x for primes p congruent to 1 modulo 4 (A002144).

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A340388 Let n = p_1*p_2*...*p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)*13^(p_2 - 1)*17^(p_3 - 1)*...*A002144(k)^(p_k - 1).

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A376428 Numbers k that occur as shorter legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

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A376429 Numbers k that occur as longer legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.

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A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).

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A102261 a(n) = A002144(n) - A002145(n).

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A113601 Intersection of A002144 and A005098.

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A145871 Smallest k such that k^2+1 is divisible by A002144(n)^7.

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A340388 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 5^(p_1 - 1)13^(p_2 - 1)17^(p_3 - 1)...*A002144(k)^(p_k - 1).