

A009003


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


65



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

Multiples of Pythagorean primes A002144 or of primitive Pythagorean triangles' hypotenuses A008846.  Lekraj Beedassy, Nov 12 2003
This is exactly the sequence of positive integers with at least one prime divisor of the form 4k + 1. Compare A072592.  John W. Layman, Mar 12 2008 and Franklin T. AdamsWatters, Apr 26 2009
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


REFERENCES

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


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
R. Chapman, Pythagorean triples and sums of squares
Steven R. Finch, LandauRamanujan Constant [Broken link]
Steven R. Finch, LandauRamanujan Constant [From the Wayback machine]
Ron Knott, Pythagorean Triples and Online Calculators
J. Pahikkala, On contraharmonic mean and Pythagorean triples, Elemente der Mathematik, 65:2 (2010), 6267.
Patrick A. Thomas, Solutions of x^5=x up to base 100
Index entries for sequences related to sums of squares


FORMULA

A005089(a(n)) > 0.  Reinhard Zumkeller, Jan 07 2013
a(n) ~ n.  Charles R Greathouse IV, Jan 13 2022


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;
end do: # R. J. Mathar, Nov 17 2014


MATHEMATICA

f[n_] := Module[{k = 1}, While[(n  k^2)^(1/2) != IntegerPart[(n  k^2)^(1/2)], k++; If[2 * k^2 >= n, k = 0; Break[]]]; k]; A009003 = {}; Do[If[f[n^2] > 0, AppendTo[A009003, n]], {n, 3, 100}]; A009003 (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)


PROG

(PARI) is_A009003(n)=setsearch(Set(factor(n)[, 1]%4), 1) \\ M. F. Hasler, May 27 2012
(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
 Reinhard Zumkeller, Jan 07 2013


CROSSREFS

Cf. A009000, A024507, A004431, A072592, A004613, A187811.
Complement of A004144. Primes in this sequence give A002144. Same as A146984 (integer contraharmonic means) as sets  see Pahikkala 2010, Theorem 5.
Cf. A083025, A084645 (primitive elements), A084646, A084647, A084648, A084649, A006339.
Sequence in context: A198389 A057100 A304436 * A071821 A334678 A201012
Adjacent sequences: A009000 A009001 A009002 * A009004 A009005 A009006


KEYWORD

nonn


AUTHOR

David W. Wilson


EXTENSIONS

Definition edited by JeanChristophe Hervé, Nov 10 2013


STATUS

approved



