A146945 -id:A146945 - cp's OEIS frontend

A120960 Pythagorean prime powers.

5, 13, 17, 25, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 125, 137, 149, 157, 169, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 289, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593

Offset: 1

Lekraj Beedassy, Jul 19 2006

nonn

1 + sum of the indices of the first two numbers in A001844 that are divisible by n, if 1 + the sum of those indices equals n. - Mats Granvik, Oct 16 2007
R. J. Turyn proved [Baliga, et al., p. 129, gives the reference] that Williamson Hadamard matrices exist for 4t = 2(p^k + 1), for all primes p such that p == 1 (mod 4). - L. Edson Jeffery, Apr 10 2012
A024362(a(n)) = 1. - Reinhard Zumkeller, Dec 02 2012

			A001844(1) = 5 is divisible by 5, A001844(3) = 25 is divisible by = 5 and 1+3+1=5, so 5 is a member.
A001844(2) = 13 is divisible by = 13, A001844(10) = 221 is divisible by = 13 and 2+10+1=13 so 13 is a member.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
A. Baliga and K. J. Horadam, Cocyclic Hadamard matrices over Z_t X Z^2_2, Australas. J. Combin. 11(1995), 123-134.
Eric Weisstein's World of Mathematics, MathWorld, Centered Square Number.

Cf. Disjoint union of A002144 and A146945.
Cf. A001844, subsequence of A000961.
Cf. A024409, subsequence of A008846.

Generate the indices with: =if(mod(1+2*row()*(row()+1);4*column()+1)=0;row();") Then sum the first two indices if it equals the column + 1. - Mats Granvik, Oct 16 2007

import Data.List (elemIndices)
a120960 n = a120960_list !! (n-1)
a120960_list = map (+ 1) $ elemIndices 1 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

A369563 Powerful numbers whose prime factors are all of the form 4*k + 1.

Original entry on oeis.org

1, 25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4225, 4913, 5329, 7225, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 21025, 21125, 22201, 24389, 24649, 28561, 29929, 32761, 34225, 36125, 37249, 38809, 42025, 48841, 50653, 52441, 54289, 54925

Offset: 1

Amiram Eldar, Jan 26 2024

nonn

Closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A004613.
Subsequence: A146945.
Similar sequence: A352492, A369564, A369565, A369566.
Cf. A002144, A008846, A175647, A334424.

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 1 && Last[#] > 1 &]; Select[Range[50000], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 1 || f[i, 2] == 1, return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) = A175647 * A334424 = 1.0654356335... .

A120961 Composite hypotenuses of primitive Pythagorean triangles.

Original entry on oeis.org

25, 65, 85, 125, 145, 169, 185, 205, 221, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 625, 629, 685, 689, 697, 725, 745, 785, 793, 841, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1145, 1157, 1165, 1189, 1205

Offset: 1

Lekraj Beedassy, Jul 19 2006

nonn

Composite entries of A008846. Disjoint union of A024409 and A146945.

Ray Chandler, Table of n, a(n) for n = 1..10000
César Aguilera, On Pythagorean Triples, hal-02151737 preprint [math.NT], 2019.

Cf. A120960, A008846, A146945, A024409.

lst = {}; Do[ If[ GCD[m, n] == 1, a = 2 m*n; b = m^2 - n^2; c = m^2 + n^2; If[ !PrimeQ@c, AppendTo[lst, c]]], {m, 3, 300}, {n, If[ OddQ@m, 2, 1], m - 1, 2}]; Take[ Union@ lst, 51] (* Robert G. Wilson v, May 02 2009 *)

Term 485, which satisfies 485^2 = 476^2 + 93^2, added by Robert G. Wilson v, May 02 2009

A159781 Values of hypotenuse of primitive Pythagorean triples which can have four different shapes (that is, four different sets of "legs").

Original entry on oeis.org

1105, 1885, 2405, 2465, 2665, 3145, 3445, 3485, 3965, 4505, 4745, 5185, 5365, 5525, 5785, 5945, 6205, 6305, 6409, 6565, 7085, 7345, 7565, 7585, 7685, 8177, 8245, 8585, 8845, 8905, 9061, 9265, 9425, 9605, 9685, 9805, 10205, 10585, 10865

Offset: 1

John T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 22 2009

nonn

This is a subsequence of A024409, which lists hypotenuse values common to more than one primitive Pythagorean triple. A024409(1) = A006278(2) = 65 is the smallest hypotenuse common to exactly two primitive Pythagorean triples; a(1) = A006278(3) = 1105 is the smallest that is common to four. [edited by Jon E. Schoenfield, Aug 19 2018]

A024362(a(n)) = 4. - Reinhard Zumkeller, Dec 02 2012

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 100 terms from Reinhard Zumkeller)

Cf. A024409 and A146945.

Cf. A006278 (8, 16, etc. shapes). - R. J. Mathar, Apr 12 2010

import Data.List (elemIndices)
a159781 n = a159781_list !! (n-1)
a159781_list = map (+ 1) $ elemIndices 4 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range[2800], f@# > 2 &] (* Robert G. Wilson v, Mar 16 2014 *)

6429 replaced by 6409 and 3 terms added by R. J. Mathar, Apr 12 2010

Missing 8585 and 8845 inserted by Reinhard Zumkeller, Dec 02 2012

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A120960 Pythagorean prime powers.

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A369563 Powerful numbers whose prime factors are all of the form 4*k + 1.

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A120961 Composite hypotenuses of primitive Pythagorean triangles.

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A159781 Values of hypotenuse of primitive Pythagorean triples which can have four different shapes (that is, four different sets of "legs").

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Haskell

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