A008846 -id:A008846 - cp's OEIS frontend

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

1, 3, 4, 6, 7, 9, 10, 13, 15, 16, 18, 21, 22, 24, 25, 27, 28, 31, 34, 36, 37, 39, 42, 43, 45, 46, 48, 49, 51, 55, 57, 58, 60, 64, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 84, 87, 88, 91, 93, 94, 97, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 120, 121, 123, 126, 127

Offset: 1

Lekraj Beedassy, May 06 2004

nonn

For the actual numbers 4n+1, see A008846(n).

Complement of A124934; A125203(a(n)) = 0; A000290 and A000217 are subsequences. - Reinhard Zumkeller, Nov 24 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

import Data.List (elemIndices)
a094178 n = a094178_list !! (n-1)
a094178_list = map (+ 1) $ elemIndices 0 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

More terms from Ray Chandler, Jun 20 2004

A120960 Pythagorean prime powers.

Original entry on oeis.org

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

A137409 Numbers that cannot be the value of 'C' in a primitive Pythagorean triple (A < B; A^2 + B^2 = C^2).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 14 2008

nonn

Complement of A008846. - R. J. Mathar, Aug 15 2010
A024362(a(n)) = 0. - Reinhard Zumkeller, Dec 02 2012
Except for the 1st term 1, complement of A004613. - Federico Provvedi, Jan 26 2019
After 1, numbers k for which A065338(k) > 1, i.e., after 1, numbers all of whose prime divisors are not of the form 4u+1. - Antti Karttunen, Dec 26 2020

			3,4,5; number 5 is not in this sequence.
5,12,13; number 13 is not in this sequence.
8,15,17; number 17 is not in this sequence.
7,24,25; number 25 is not in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004613, A008846, A020882, A024362, A065338, A137407.

Subsequences: A125667 (the odd terms), A339875.

import Data.List (elemIndices)
a137409 n = a137409_list !! (n-1)
a137409_list = map (+ 1) $ elemIndices 0 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

okQ[1] = True;
okQ[n_] := AnyTrue[FactorInteger[n][[All, 1]], Mod[#, 4] != 1&];
Select[Range[100], okQ] (* Jean-François Alcover, Mar 10 2019, after Federico Provvedi's comment *)

A065338(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f); };
isA137409(n) = ((1==n)||(A065338(n)>1)); \\ Antti Karttunen, Dec 26 2020

Extended by R. J. Mathar, Aug 15 2010

A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

5, 13, 25, 17, 41, 61, 37, 85, 113, 65, 145, 181, 29, 101, 221, 265, 145, 313, 365, 53, 197, 421, 481, 257, 65, 545, 613, 85, 325, 685, 89, 761, 401, 841, 925, 125, 485, 1013, 1105, 73, 577, 1201, 149, 1301, 173, 677, 1405, 1513, 785, 185, 1625, 1741, 109, 229

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of C values is A020882(n) (allowing repetitions) and A008846(n) (excluding repetitions).

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=5, a(2)=13, a(3)=25 and a(4)=17.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A020882, A008846, A156678, A156682.

Cf. A037213.

a156679 n = a156679_list !! (n-1)
a156679_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = w : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iHarvey P. Dale, May 10 2020 *)

A146945 Hypotenuses of primitive Pythagorean triples which are not prime numbers and which are the hypotenuse of a unique triangle.

Original entry on oeis.org

25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4913, 5329, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 22201, 24389, 24649, 28561, 29929, 32761, 37249, 38809, 50653, 52441, 54289, 58081, 66049, 68921, 72361, 76729, 78125

Offset: 1

John Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 20 2009

nonn

Each term is a prime power of the form p^e where p is in A002144 and e>1.
A proper subset of A120960 by eliminating A002144.
A proper subset of A120961 by eliminating A024409.
A proper subset of A008846 by eliminating A002144 and A024409.
A proper subset of A020882 by eliminating A002144, A024409 and duplicate entries.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A020882, A008846, A002144, A024409, A120960, A120961.

lst1 = {1, 1}; lst2 = {}; Do[ If[ GCD[m, n] == 1, a = 2m*n; b = m^2 - n^2; c = m^2 + n^2; If[ !PrimeQ@c, AppendTo[lst1, c]]], {m, 3, 1000}, {n, If[OddQ@m, 2, 1], m - 1, 2}]; lst1 = Sort@ lst1; Do[ If[ lst1[[n - 1]] != lst1[[n]] && lst1[[n]] != lst1[[n + 1]], AppendTo[lst2, lst1[[n]]]], {n, 2, Length@ lst1 - 1}]; Take[lst2, 50] (* Robert G. Wilson v, May 02 2009 *)

a(7) corrected by and a(17) and further terms from Robert G. Wilson v, May 02 2009

Minor edits to comments. - Ray Chandler, Nov 27 2019

A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, 17, 9, 40, 41, 11, 60, 61, 12, 35, 37, 13, 84, 85, 15, 112, 113, 16, 63, 65, 17, 144, 145, 19, 180, 181, 20, 21, 29, 20, 99, 101, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53

Offset: 1

Colin Barker, Nov 20 2015

a(3*k+1)*a(3*k+2) / (a(3*k+1)+a(3*k+2)+a(3*k+3)) is always an integer for k >= 0. Also note that a(3*k+1)*a(3*k+2)/2 is never a perfect square. - Altug Alkan, Apr 08 2016

			The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.

Colin Barker, Table of n, a(n) for n = 1..10000
Yoshinosuke Hirakawa and Hideki Matsumura, A unique pair of triangles, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.
Eric Weisstein's World of Mathematics, Pythagorean Triple
Wikipedia, Pythagorean triple

Cf. A008846, A020882, A046086, A046087, A103606, A139794, A156678, A156679.

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... .

A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.

Original entry on oeis.org

1, 3, 6, 10, 15, 16, 21, 22, 24, 28, 36, 37, 39, 42, 45, 46, 48, 51, 55, 58, 66, 67, 69, 72, 76, 78, 79, 84, 88, 91, 94, 97, 105, 106, 108, 111, 115, 120, 121, 123, 126, 130, 135, 136, 139, 142, 153, 154, 157, 163, 168, 171, 172, 174, 177, 181, 186, 190, 192, 193

Offset: 1

Hugo Pfoertner, Sep 13 2024

nonn

Sorted distinct values of ({A081961} - 1)/4.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A008846, A081961, A376208.

\\ Uses function is_a376208 from A376208
is_a376208(n,1)

from math import gcd, isqrt
test_all_k_upto   = 193
A375750, x, limit = set(), 2, test_all_k_upto * 4 + 1
while x**2 + (lowY := isqrt(2*x**2)-x)**2 < limit:
   for y in range(min(x-1,(yy:=isqrt(limit - x**2))-(yy%2 == x%2)), lowY,-2):
       if gcd(x,y) == 1: A375750.add(((x**2 + y**2) - 1) // 4)
   x += 1
print(A375750:=sorted(A375750)) # Karl-Heinz Hofmann, Sep 17 2024

A376208 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an even short leg.

Original entry on oeis.org

4, 7, 9, 13, 16, 18, 21, 25, 27, 31, 34, 36, 43, 46, 49, 51, 55, 57, 60, 64, 66, 70, 73, 76, 81, 87, 91, 93, 94, 99, 100, 102, 111, 112, 114, 121, 123, 126, 127, 133, 136, 141, 144, 148, 150, 156, 157, 160, 165, 169, 171, 172, 175, 181, 183, 186, 189, 196, 198, 202

Offset: 1

Hugo Pfoertner, Sep 20 2024

nonn

This sequence is not A094178 \ A375750, because there are hypotenuses for which both kinds of triangles exist. The smallest example occurs for hypotenuse c = 4*a(5) + 1 = 65. The triangle (16, 63, 65) has an even short leg, but there is also the triangle (33, 56, 65) with an odd short leg. Thus, 16 = (65-1)/4 is a term in this sequence and in A375750.

Sorted distinct values of ({A081985} - 1)/4.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

{a(n)} U A375750 = A094178.

Cf. A004613, A008846, A081985.

is_a376208(n,r=0) = my(c=4*n+1, q=qfbsolve(Qfb(1,0,1), c^2, 3), qd=#q, is=0); for(k=1, qd-1, if(vecmin(abs(q[k]))%2==r && gcd([c,q[k]])==1, is=1; break)); is

# for an array from the beginning
from math import gcd, isqrt
test_all_k_upto = 202
A376208, limit = set(), test_all_k_upto * 4 + 1
for x in range(2,isqrt(limit)+1):
    for y in range(min(((d:=isqrt(2*x**2)-x))-(d%2==x%2), (yy:=isqrt(limit-x**2))-(yy%2==x%2)),0,-2):
        if gcd(x, y) == 1: A376208.add((x**2 + y**2 - 1) // 4)
print(A376208:=sorted(A376208)) # Karl-Heinz Hofmann, Sep 28 2024

# for testing high single terms
from math import isqrt, gcd
from sympy import factorint
def A376208_isok(k):
    c  = k * 4 + 1
    if any([(pf-1) % 4 for pf in factorint(c)]): return False # (Test imported from A008846)
    y2 = c - (x2:=(x:=isqrt(c))**2)
    while 2*x*(y:=isqrt(y2)) < x2-y2:
        if y2 == y**2 and gcd(x, y) == 1: return True
        x -= 1
        y2 = c - (x2:=x**2) # Karl-Heinz Hofmann, Oct 17 2024

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

cp's OEIS Frontend

A094178 Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).

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A120960 Pythagorean prime powers.

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A137409 Numbers that cannot be the value of 'C' in a primitive Pythagorean triple (A < B; A^2 + B^2 = C^2).

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A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A146945 Hypotenuses of primitive Pythagorean triples which are not prime numbers and which are the hypotenuse of a unique triangle.

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A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

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

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A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.