

A009003


Hypotenuse numbers (squares are sums of 2 nonzero squares).


69



5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 140
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OFFSET

1,1


COMMENTS

Circumradius R of the triangles such that the area, the sides and R are integers.  Michel Lagneau, Mar 03 2012
The 2 squares summing to a(n)^2 cannot be equal because sqrt(2) is not rational.  JeanChristophe Hervé, Nov 10 2013
Closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 4k + 1 with multiplicity one, which are also those for which there exists a unique integer triangle = A084645.  JeanChristophe Hervé, Nov 11 2013
a(n) are numbers whose square is the mean of two distinct nonzero squares. This creates 1to1 mapping between a Pythagorean triple and a "Mean" triple. If the Pythagorean triple is written, abnormally, as {j, k, h} where j^2 +(j+k)^2 = h^2, and h = a(n), then the corresponding "Mean" triple with the same h is {k, 2j, h} where (k^2 + (k+2j)^2)/2 = h^2. For example for h = 5, the Pythagorean triple is {3, 1, 5} and the Mean triple is {1, 6, 5}.  Richard R. Forberg, Mar 01 2015
Integral side lengths of rhombuses with integral diagonals p and q (therefore also with integral areas A because A = pq/2 is some multiple of 24). No such rhombuses are squares.  Rick L. Shepherd, Apr 09 2017
Conjecture: these are bases n in which exists an nadic integer x satisfying x^5 = x, and 5 is the smallest k>1 such that x^k =x (so x^2, x^3 and x^4 are not x). Example: the 10adic integer x = ...499879186432 (A120817) satisfies x^5 = x, and x^2, x^3, and x^4 are not x, so 10 is in this sequence. See also A120817, A210850 and A331548.  Patrick A. Thomas, Mar 01 2020
Didactic comment: When students solve a quadratic equation a*x^2 + b*x + c = 0 (a, b, c: integers) with the solution formula, they often make the mistake of calculating b^2 + 4*a*c instead of b^2  4*a*c (especially if a or c is negative). If the root then turns out to be an integer, they feel safe. This sequence lists the absolute values of b for which this error can happen. Reasoning: With p^2 = b^2  4*a*c and q^2 = b^2 + 4*a*c it follows by addition immediately that p^2 + q^2 = 2*b^2. If 4*a*c < 0, let p = x + y and q = x  y. If 4*a*c > 0, let p = x  y and q = x + y. In both cases follows that y^2 + x^2 = b^2. So every Pythagorean triple gives an absolute value of b for which this error can occur. Example: From (y, x, b) = (3, 4, 5) follows (q^2, b^2, p^2) = (1, 25, 49) or (p^2, b^2, q^2) = (1, 25, 49) with abs(4*a*c) = 24.  Felix Huber, Jul 22 2023


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98104.


LINKS



FORMULA



MAPLE

isA009003 := proc(n)
local p;
for p in numtheory[factorset](n) do
if modp(p, 4) = 1 then
return true;
end if;
end do:
false;
end proc:
for n from 1 to 200 do
if isA009003(n) then
printf("%d, ", n) ;
end if;


MATHEMATICA

Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)


PROG

(PARI) list(lim)=my(v=List(), u=vectorsmall(lim\=1)); forprimestep(p=5, lim, 4, forstep(n=p, lim, p, u[n]=1)); for(i=5, lim, if(u[i], listput(v, i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022
(Haskell)
import Data.List (findIndices)
a009003 n = a009003_list !! (n1)
a009003_list = map (+ 1) $ findIndices (> 0) a005089_list
(Python)
from itertools import count, islice
from sympy import primefactors
def A009003_gen(): # generator of terms
return filter(lambda n:any(map(lambda p: p % 4 == 1, primefactors(n))), count(1))


CROSSREFS

Complement of A004144. Primes in this sequence give A002144. Same as A146984 (integer contraharmonic means) as sets  see Pahikkala 2010, Theorem 5.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



