A008846 -id:A008846 - cp's OEIS frontend

A159935 Least integer such that a(n)^2 - n is the sum of two nonzero squares.

5, 3, 2, 4, 3, 5, 4, 3, 4, 7, 6, 4, 5, 9, 4, 5, 6, 5, 6, 6, 5, 11, 12, 5, 7, 15, 6, 8, 6, 7, 8, 6, 7, 13, 6, 8, 7, 21, 8, 7, 9, 7, 10, 12, 7, 15, 8, 7, 10, 9, 10, 8, 9, 11, 8, 9, 8, 17, 30, 8, 10, 9, 8, 9, 9, 13, 10, 18, 9, 11, 12, 9, 12, 9, 10, 10, 9, 15, 16, 9, 10, 11, 10, 10, 11, 21, 12, 10, 15

Offset: 0

Franklin T. Adams-Watters, Apr 26 2009

nonn

Cf. A008846, A050795.

issum2sq(n) = local(fm, hf); hf=0;fm=factor(n);for(i=1,matsize(fm)[1],if(fm[i,1]==2,if(fm[i,2]%2,hf=1),if(fm[i,1]%4==1,hf=1,if(fm[i, 2]%2,return(0)))));hf
minsum2sq(n) = local(k); k=1;while(!issum2sq(k^2-n),k++);k
/* Note: the issum2sq function depends on PARI returning -1 as a factor for negative n. */

A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

Original entry on oeis.org

151, 463, 571, 631, 643, 991, 1063, 1171, 1831, 2083, 2311, 4951, 5023, 6211, 6703, 6763, 7723, 7951, 9043, 11383, 12163, 12391, 13183, 14851, 15031, 17431, 19231, 19543, 20143, 22051, 23143, 25951, 26371, 27283, 28351, 29131, 30643, 32803

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

nonn

151*75-4=11321 (prime), 151*75+4=11329 (prime), ..

Subsequence of A068229. Cf. A008846, A020882, A068229, A086519, A158708, A164620, A164621

lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]-4]&&PrimeQ[p*Floor[p/2]+4],AppendTo[lst,p]],{n,8!}];lst

Edited by Charles R Greathouse IV, Nov 02 2009

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

A174825 Prime concatenations p = concatenation of c, b, and a where a, b, c is a primitive Pythagorean triple, a < b < c.

Original entry on oeis.org

25247, 18517657, 42541687, 48148031, 305224207, 461380261, 929920129, 1249960799, 4141414091, 13811020931, 17451736177, 18011680649, 19011820549, 22852204603, 25812460781, 27492580949, 39653956267, 47094700291

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 30 2010

c^2 = b^2 + a^2 with c > b > a relatively prime, i.e. a primitive Pythagorean triple
Note two curiosities for 6th term p(6) = cat(461, 280, 261) = prime(24423734):
cat(261, 380, 461): 261380461 = prime(14267135) also prime, SMALLEST of this type
Additionally p(6) is also the FIRST such concatenation with a prime hypotenuse: 461 = prime(89) Same is true for p(8) = 1249960799 = prime(62841771), 7999601249 = prime(367766086), 1249 = prime(204)
p(15) = 25812460781 = prime(1125896092), 78124602581 = prime(3250321954)
but hypotenuse 2581 = 29 * 89 and short leg 781 = 11 * 71 are both composite
p(18) = 47094700291= prime(2001581081), 29147004709 = prime(1264629019), 4709 = 17 * 277, 291 = 3 * 97

			25^2 = 24^2 + 7^2, and cat(25, 24, 7) = 25247 is prime, so 25247 is in the sequence.
5^2 = 4^2 + 3^2, but cat(5, 4, 3) = 543 = 3*181 is not prime, so 543 is not in the sequence.

W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, New York: Dover, 1987
L. E. Dickson: "Rational Right Triangles", ch. 4 in History of the Theory of numbers, vol. II, Dover Publications 2005
W. Sierpinski: Pythagorean Triangles, Mineola, NY, Dover Publications, Inc, 2003

Cf. A020882, A008846

Edited by Franklin T. Adams-Watters, Aug 27 2012

A211176 Numbers n which are the hypotenuse of a Pythagorean triple with n' as a leg, where n' is the arithmetic derivative of n.

Original entry on oeis.org

125, 625, 23125, 142805, 210125, 371293, 7983625, 9370805, 25757525, 50062025, 120670225, 489766225, 881052625, 1471596725, 2307267625, 2489771125, 3145529225, 3474871553, 6975757441, 7977558641

Offset: 1

Paolo P. Lava, Feb 01 2013

This sequence is a subsequence of A008846. - Ray Chandler, Jan 27 2017

			n = 23125, n' = 19125 and sqrt(n^2-n'^2) = 13000.

Cf. A002144, A002365, A002366, A003415, A008846, A009003, A009004, A210503.

with(numtheory); ListA211176:= proc(q)local a,n,p;
for n from 2 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if n<>a and type(sqrt(n^2-a^2),integer) then print(n); fi;
od; end: ListA211176(10^9);

A002144(n)^A002365(n) and A002144(n)^A002366(n) are terms of the sequence for all n. - Ray Chandler, Jan 27 2017

Name and Maple program corrected by Paolo P. Lava, Sep 30 2013
a(12)-a(16) from Donovan Johnson, Sep 30 2013
a(17)-a(18) from Ray Chandler, Jan 25 2017
a(19)-a(20) from Ray Chandler, Jan 27 2017

A277534 Least hypotenuse, c, of a Primitive Pythagorean Triangle (PPT) such that the difference between it, c, and its greater leg, b, is n; or 0 if no such PPT exists.

Original entry on oeis.org

5, 17, 0, 0, 65, 0, 0, 29, 65, 185, 0, 0, 169, 0, 0, 0, 221, 333, 0, 0, 273, 0, 0, 0, 157, 481, 0, 0, 1189, 0, 0, 641, 1353, 629, 0, 0, 1517, 0, 0, 425, 1681, 777, 0, 0, 1845, 0, 0, 0, 205, 925, 0, 0, 2173, 0, 0, 0, 2337, 1073, 0, 0, 2501, 0, 0, 0, 2665, 1221, 0, 0, 2829, 0, 0, 1405, 2993, 1369, 0

Offset: 1

Ron Knott and Robert G. Wilson v, Jun 05 2014

n = 1, 2, 5, 8, 9, 10, 13, 17, 18, 21, 25, ..., satisfies the first criterion;
a(n) = 0 for n = 3, 4, 6, 7, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, ..., ;
a(n) = 0 for 5832 of the first 10000 terms;
a(8n) = 0 for 832 of the first 10000 terms;
a(8n) = 0 for n: 2, 3, 6, 7, 8, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, ..., ;
a(8n+1) > 0;
a(8n+2) > 0; a linear 2nd-order recurrence: a(n) = 2*a(n-1) - a(n-2) with a(1) = 185 & a(2) = 333;
a(8n+3) = 0;
a(8n+4) = 0;
a(8n+5) > 0;
a(8n+6) = 0;
a(8n+7) = 0;
Prime terms: 5, 17, 29, 157, 641, 3821, 4201, 17749, 21601, 31981, 38273, 44789, 61129, 66173, 72161, 100673, 108541, 114553, 121421, 142973, 165541, 173777, 182141, 204733, 213881, 225889, 235493, 281837, ..., .

			a(1) is 5 since the PPT (3,4,5) satisfies the first stated criterion; a(2) is 17 since the PPT (8,15,17) satisfies the first stated criterion; a(3) = 0 since there exists no PPT that satisfies the stated criteria; etc.

Ron Knott and Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A008846, A020882, A242219.

f[n_] := FindInstance[ a^2 + b^2 == c^2 && Mod[c, 4] == 1 && 0 < a < b < c && c - b == n, {a, b, c}, Integers][[1, 3, 2, 1, 1, 3]] + 1 /. 1 + {}[[1, 3, 2, 1, 1, 3]] -> 0; f[1] = 5; Array[f, 75]

A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.

Original entry on oeis.org

11, 46, 52, 117, 142, 198, 236, 286, 415, 488, 524, 549, 621, 666, 835, 873, 908, 970, 1001, 1199, 1388, 1432, 1692, 1757, 1962, 1964, 1971, 2035, 2041, 2366, 2392, 2630, 2655, 2681, 2702, 2815, 2826, 3195, 3421, 3544, 3664, 3715, 4048, 4070, 4097, 4356

Offset: 1

R. J. Mathar, Feb 06 2017

nonn

If x and y are coprime, so obviously are also (x,z) and (y,z).
The ordered values of the bases of the cubes, z, are a subsequence of (and conjecturally the same as) A008846.
For production purposes we advice to use the parametrized representations (see references).

			2^2 + 11^2 = 5^3, so 11 is in the sequence.
9^2 + 46^2 = 13^3, so 46 is in the sequence.
47^2 + 52^2 = 17^3, so 52 is in the sequence.
44^2 + 117^2 = 25^2, so 117 is in the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1172
Imin Chen, On the equation s^2+y^(2p)=alpha^3, Math. Comp. 77 (262) (2008) 1223-1227.
Sander R. Dahmen, A refined modular approach to the diophantine equation x^2+y^(2n)=z^3, arXiv:1002.0020 [math.NT] (2010).

Subsequence of A282093. Cf. A099533.

# slow version for demonstration only.
isA282095 := proc(y)
    local x,z3 ;
    for x from 1 to y do
        if igcd(x,y) = 1 then
            z3 := x^2+y^2 ;
            if isA000578(z3) then
                return true ;
            end if;
        end if;
    end do:
    return false ;
end proc:
for y from 1 do
    if isA282095(y) then
        printf("%d,\n",y) ;
    end if;
end do:

okQ[y_] := Module[{x, z3}, For[x=1, xJean-François Alcover, Dec 04 2017, after R. J. Mathar *)

{y: x^2 + y^2 = z^3; gcd(x,y) = 1; 1 <= x <= y; x, y, z in N}

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

Original entry on oeis.org

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

Offset: 1

Mo Li, Oct 17 2019

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = log(2)/Pi^2 = 0.07023049... See A118858.

			For n=90, the triples are
   {3,  4,  5},  3 +  4 +  5 = 12 < 90
   {5, 12, 13},  5 + 12 + 13 = 30 < 90
   {7, 24, 25},  7 + 24 + 25 = 56 < 90
   {8, 15, 17},  8 + 15 + 17 = 40 < 90
   {9, 40, 41},  9 + 40 + 41 = 90
  {12, 35, 37}, 12 + 35 + 37 = 84 < 90
  {20, 21, 29}, 20 + 21 + 29 = 70 < 90
so a(90)=7.

Ron Knott, Pythagorean Triples and Online Calculators
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A008846, A024364, A118858, A249750.

A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

Original entry on oeis.org

221, 325, 377, 425, 493, 629, 697, 725, 925, 1025, 1073, 1189, 1325, 1517, 1537, 1769, 1885, 1961, 2173, 2257, 2405, 2501, 2665, 2701, 2993, 3145, 3233, 3293, 3445, 3485, 3649, 3869, 3965, 3977, 4453, 4505, 4717, 4745, 5141, 5185, 5353, 5429, 5777, 5785, 5917

Offset: 1

Piotr Lipski, Aug 16 2024

nonn

			221 is a term since the following primitive Pythagorean triples have 221 in first, second and third position: (221, 24420, 24421), (60, 221, 229), (21, 220, 221).

Mathematics Stack Exchange, Is there an integer that serves as the short leg of a primitive Pythagorean triple, the long leg of another, and the hypotenuse of a third?
Rémy Sigrist, PARI program

Intersection of A008846, A024352 and A024354.

Cf. A263728.

PARI
```
\\ See Links section.
```

a(n) == 1 (mod 4). - Hugo Pfoertner, Aug 18 2024

More terms from Rémy Sigrist, Aug 17 2024

A157075 Positive integers n for which the Diophantine equation x^2 + y^2 = n^2/2 has relatively prime solutions.

Original entry on oeis.org

10, 26, 34, 50, 58, 74, 82, 106, 122, 130, 146, 170, 178, 194, 202, 218, 226, 250, 274, 290, 298, 314, 338, 346, 362, 370, 386, 394, 410, 442, 458, 466, 482, 514, 530, 538, 554, 562, 578, 586, 610, 626, 634, 650, 674, 698, 706, 730, 746, 754, 778, 794, 802

Offset: 1

Knut Angstrom (angstrom.knut(AT)telia.com), Feb 22 2009

nonn

a(n) = 2 * A008846(n). [From N. J. A. Sloane, Feb 24 2009]

Edited and extended by Max Alekseyev, Apr 22 2010

cp's OEIS Frontend

A159935 Least integer such that a(n)^2 - n is the sum of two nonzero squares.

Views

Author

Keywords

Crossrefs

Programs

PARI

A164622 Primes p such that p*floor(p/2) - 4 and p*floor(p/2) + 4 are prime numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A174825 Prime concatenations p = concatenation of c, b, and a where a, b, c is a primitive Pythagorean triple, a < b < c.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Extensions

A211176 Numbers n which are the hypotenuse of a Pythagorean triple with n' as a leg, where n' is the arithmetic derivative of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A277534 Least hypotenuse, c, of a Primitive Pythagorean Triangle (PPT) such that the difference between it, c, and its greater leg, b, is n; or 0 if no such PPT exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

A164622 Primes p such that pfloor(p/2) - 4 and pfloor(p/2) + 4 are prime numbers.

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.