A009000 -id:A009000 - cp's OEIS frontend

A024507 Numbers that are the sum of 2 distinct nonzero squares (with repetition).

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 65, 68, 73, 74, 80, 82, 85, 85, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 125, 130, 130, 136, 137, 145, 145, 146, 148, 149, 153, 157, 160, 164, 169, 170, 170, 173, 178, 180, 181, 185, 185, 193

Offset: 1

Clark Kimberling

nonn

Cf. A009000, A009003, A024507, A004431 (duplicates removed), A055096.

nn=10000;A024507=Table[x^2+y^2,{x,Sqrt[nn]},{y,x+1,Sqrt[nn-x^2]}]//Flatten//Sort (* Zak Seidov, Apr 07 2011*)

Name edited by Zak Seidov, Apr 08 2011

A344330 Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 15, 20, 30, 40, 45, 50, 60, 65, 68, 70, 75, 78, 80, 90, 100, 105, 110, 120, 130, 135, 136, 140, 150, 156, 160, 165, 170, 175, 180, 190, 195, 200, 204, 210, 220, 222, 225, 230, 234, 240, 250, 255, 260, 270, 272, 280, 285, 290, 300, 310, 312, 315, 320, 325, 330, 340, 345, 350, 360, 369, 370

Offset: 1

Bernard Schott, May 15 2021

nonn

This sequence is a generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008.
Some notations: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
Side s of such tiled squares must satisfy the Diophantine equation s^2 = z * (a^2+b^2).
There are two types of solutions. See A344331 for type 1 and A344332 for type 2.
If q is a term, k * q is another term for k > 1.

			-> Example of type 1:
Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
.
-> Example of type 2:
Square 15 x 15 with a = 3, b = 4, s = 15, z = 9.
      ________ ________ ________ _____
     |        |        |        |     |
     |        |        |        |     |
     |        |        |        |_____|
     |_______ |________|________|     |
     |        |        |        |     |
     |        |        |        |_____|
     |        |        |        |     |
     |________|________|________|     |
     |        |        |        |_____|
     |        |        |        |     |
     |        |        |        |     |
     |_____ __|___ ____|_ ______|_____|
     |     |      |      |      |     |
     |     |      |      |      |     |
     |_____|______|______|______|_____|
Remarks:
- With terms as 10, 20, ... we only obtain sides of squares of type 1:
10 is a term of this type because the square 10 X 10 only can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2 (see first example),
20 is another term of this type because the square 20 X 20 only can be tiled with 80 squares of size 1 x 1 and 80 squares of size 2 x 2.
- With terms as 15, 65, ... we only obtain sides of squares of type 2:
15 is a term of this type because the square 15 X 15 only can be tiled with 9 squares of size 3 X 3 and 9 squares of size 4 X 4 (see second example),
65 is another term of this type because the square 65 X 65 only can be tiled with 25 squares of size 5 X 5 and 25 squares of size 12 X 12.
- With terms as 30, 60, ... we obtain both sides of squares of type 1 and of type 2:
30 is a term of type 1 because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2, but,
30 is also a term of type 2 because the square 30 X 30 can be tiled with 9 squares of size 6 X 6 and 9 squares of size 8 X 8.

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Subsequences: A008592 \ {0}, A008597 \ {0}, A034262 \ {0,1}.

Cf. A344331, A344332, A344333, A344334.

pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000
isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);
isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }
isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (dx*y*(x^2+y^2), [1..m]), s);}
isok1(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Jun 04 2021

Corrected by Michel Marcus, May 18 2021

Incorrect term 145 removed by Michel Marcus, Jun 04 2021

A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.

Original entry on oeis.org

25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, 325, 338, 340, 350, 365, 370, 375, 377, 390, 400, 410, 425, 435, 442, 445, 450, 455, 475, 481, 485, 493, 500, 505, 507, 510, 520, 525

Offset: 1

David W. Wilson

nonn

Also, hypotenuses of Pythagorean triangles in Pythagorean triples (a, b, c, a < b < c) such that a and b are the hypotenuses of Pythagorean triangles, where the Pythagorean triples (x1, y1, a) and (x2, y2, b) are similar triangles. Sequence gives c values. - Naohiro Nomoto
Any multiple of a term of this sequence is also a term. The primitive elements are the products of two primes, not necessarily distinct, that are == 1 (mod 4): A121387. - Franklin T. Adams-Watters, Dec 21 2015
Numbers appearing more than once in A009000. - Sean A. Irvine, Apr 20 2018

			25^2 = 24^2 + 7^2 = 20^2 + 15^2.
E.g., (a = 15, b = 20, c = 25, a^2 + b^2=c^2); 15 and 20 are the hypotenuses of Pythagorean triangles. The Pythagorean triples (9, 12, 15) and (12, 16, 20) are similar triangles. So c = 25 is in the sequence. - _Naohiro Nomoto_

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004431, A009000, A118882, A121387.

filter:= proc(n) add(`if` (t[1] mod 4 = 1, t[2],0), t = ifactors(n)[2]) >= 2 end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 21 2015

f[n_] := Module[{i = 0, k = 0}, Do[If[Sqrt[n^2 - i^2] == IntegerPart[Sqrt[n^2 - i^2]], k++], {i, n - 1, 1, -1}]; k];
lst = {}; Do[If[f[n] > 2, AppendTo[lst, n]], {n, 4*5!}];
lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Of the form b(i)*b(j)*k, where b(n) is A004431(n). Numbers whose prime factorization includes at least 2 (not necessarily distinct) primes congruent to 1 mod 4. - Franklin T. Adams-Watters, May 03 2006. [Typo corrected by Ant King, Jul 17 2008]

A344332 Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

15, 30, 45, 60, 65, 75, 90, 105, 120, 130, 135, 150, 165, 175, 180, 195, 210, 225, 240, 255, 260, 270, 285, 300, 315, 325, 330, 345, 350, 360, 369, 375, 390, 405, 420, 435, 450, 455, 465, 480, 495, 510, 520, 525, 540, 555, 570, 585, 600, 615, 630, 645, 650, 660, 671, 675, 690, 700, 705, 715, 720, 735, 738, 750, 765, 780, 795, 810, 825, 840, 845, 855, 870, 875, 885, 900

Offset: 1

Bernard Schott, May 20 2021

nonn

This sequence is relative to the generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008 (see A344330).
There are two types of solutions, the second one is proposed here, while type 1 is described in A344331.
If m is a term and k > 1, k * m is another term.
Every term (primitive or not primitive) is the side of an elementary square of type 2 (see A346263).
Some notations: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
-> Primitive squares
Side s of primitive squares of type 2 must satisfy the Diophantine equation s^2 = z * (a^2+b^2) with the conditions a^2+b^2 = c^2 and gcd(a, b, c) = 1.
In this case, q = a/(c-b) must be odd, and side s = q*c = a*c/(c-b) = (a+b)*c/a with a number of squares z = q^2 = (a/(c-b))^2 = ((b+c)/a)^2.
Indeed, these conditions give exactly the following solutions for n >= 2: s = n^4-(n-1)^4 (A005917), a = 2*n-1 (A005408), b = 2*n*(n-1) (A046092), c = 2*n*(n-1)+1 (A001844), z = (2*n-1)^2 (A016754); this results come from the identity:
   (n^4 - (n-1)^4)^2 = (2*n-1)^2 * ((2*n-1)^2 + (2*n*(n-1))^2).
For n >= 2, every primitive square is composed by a square S1 of z = (2*n-1)^2 large squares with side b = 2*n*(n-1), then an edge on two sides of this square S1 of z = (2*n-1)^2 small squares with side a = 2*n-1.
See example with design of square of side s = 15 with a = 3, b = 4, c = 5, q = 3, z = 9, obtained with n= 2.
-> Non-primitive squares
If s is the side of a primitive square of type 2, then every k * s, k > 1 is a non-primitive term that gives two distinct tilings of type 2.
The square ks X ks can be tiled with z = q^2 = (2n-1)^2 = (a/(c-b))^2 = ((b+c)/a)^2 squares of side ka and of side kb, but also,
The square ks X ks can be tiled with z = k^2*q^2 = ((2n-1)*k)^2 = (k*a/(c-b))^2 = (k*(b+c)/a)^2 squares of side a and of side b (see example).

			Primitive square with s = 15:
  a = 3, b = 4, c = 5, s = 15, z = 9; s = 15 is the side of primitive square, with  z = 9 squares of size 3 x 3 and 9 squares of size 4 x 4
Non-primitive square k*s = 2*15 = 30:
  a = 3, b = 4, c = 5, s = 30, z = 36, this square is obtained with 4 copies of the primitive square as below.
  a = 6, b = 8, c = 10, s = 30, z = 9, this square and its tiling are exactly as the primitive square with scale 2.
               b = 4 (or = 8)     a = 3 (or = 6)
            ________ ________ ________ ______ ______________________________
           |        |        |        |      |                              |
           |        |        |        |      |                              |
           |        |        |        |______|                              |
           |_______ |________|________|      |                              |
           |        |        |        |      |                              |
           |        |        |        |______|                              |
           |        |        |        |      |                              |
           |________|________|________|      |                              |
           |        |        |        |______|                              |
           |        |        |        |      |                              |
           |        |        |        |      |                              |
           |_____ __|___ ____|_ ______|______|                              |
           |     |      |      |      |      |                              |
           |     |      |      |      |      |                              |
           |_____|______|______|______|______|______________________________|
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |                                 |                              |
           |_________________________________|______________________________|
                      s = 15               s = 30

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Cf. A001844, A005408, A005917, A046092, A046754, A344330, A344331.

pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000
isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);
isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }
isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Jun 04 2021

A057100 Hypotenuses of Pythagorean triangles (ordered by the product of the sides).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 25, 29, 30, 41, 37, 34, 35, 39, 40, 50, 61, 45, 52, 51, 50, 65, 53, 55, 85, 58, 65, 60, 75, 82, 65, 74, 68, 65, 70, 78, 113, 73, 101, 75, 85, 80, 85, 91, 100, 89, 85, 122, 87, 90, 145, 123, 104, 95, 111, 102, 97, 100, 145, 130, 125, 106

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

			a(1)=5 since 3*4*5=60 is smallest possible positive product

Cf. A009000, A057096.

maxShortLeg = 66; terms = 64;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 3]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057098(n)*A057099(n)) =sqrt(A057098(n)^2+A057099(n)^2)

A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

Original entry on oeis.org

6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850

Offset: 1

Jon Perry, Jan 13 2004

Intersection of triangular numbers with sumset of triangular numbers. Triangular number analog of what for squares is {A057100(n)^2} = {A009000(n)^2}. {A000217} INTERSECT {A000217 + A000217}. - Jonathan Vos Post, Mar 09 2007

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.

Michael S. Branicky, Table of n, a(n) for n = 1..11915

Cf. A000217, A012132, A051533, A057100.

trn[i_]:=Module[{trnos=Accumulate[Range[i]],t2s},t2s=Union[Total/@ Tuples[ trnos,2]];Intersection[trnos,t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)

t(i) = i*(i+1)/2;
{ v=vector(100,i,t(i)); y=vector(100); c=0; for (i=1,30, for (j=i,30, x=t(i)+t(j); f=0; for (k=1,100,if (x==v[k],f=1;break)); if (f==1,y[c++ ]=x))); select(x->(x>0), vecsort(y,,8)) } \\ slightly edited by Michel Marcus, Apr 15 2021

lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break;)););} \\ Michel Marcus, Apr 15 2021

from itertools import count, takewhile
def aupto(lim):
    t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
    s = set(a+b for i, a in enumerate(t) for b in t[i:])
    return sorted(s & set(t))
print(aupto(3000)) # Michael S. Branicky, Jun 21 2021

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. - Max Alekseyev, Oct 24 2008

a(n) = A000217(A012132(n)). - Ivan N. Ianakiev, Jan 17 2013

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005

A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

Original entry on oeis.org

1, 2, 1, 6, 12, 27, 19, 31, 64, 48, 96, 72, 1, 26, 28, 15, 3, 26, 24, 24, 48, 64, 44, 35, 48, 96, 108, 151, 131, 223, 447, 831, 639, 190, 380, 299, 507, 663, 1212, 904, 209, 7, 36, 104, 17, 87, 116, 211, 264, 264, 165, 103, 143, 151, 204, 303, 536, 1055, 860, 1688, 3156, 2592, 1341, 1399

Offset: 0

Scott R. Shannon, Sep 08 2021

In the first one million terms the largest value is a(987016) = 123592518669. In this range the smallest number that has not yet appeared is 9.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 1 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 1 = 9 = 3^2 is the next smallest square.
a(60) = 3156 as a(59)^2 + 60^2 = 2849344 + 3600 = 2852944, and 2852944 + 3156 = 2856100 = 1690^2 is the next smallest square.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000290, A103605, A009000, A005408, A347754.

Nest[Append[#, Block[{k = 1, m = Last[#1]}, While[! IntegerQ@ Sqrt[#2^2 + m^2 + k], k++]; k]] & @@ {#, Length@ #} &, {1}, 63] (* Michael De Vlieger, Sep 08 2021 *)

lista(nn) = {my(prec = 1, list=List()); listput(list, prec); for (n=1, nn, my(k = 1); while (!issquare(prec^2+n^2+k), k++); listput(list, k); prec = k;); Vec(list);} \\ Michel Marcus, Sep 13 2021

from math import isqrt
A347594_list = [1]
for n in range(1,10**3):
    m = A347594_list[n-1]**2+n**2
    A347594_list.append((isqrt(m)+1)**2-m) # Chai Wah Wu, Sep 12 2021

A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

Original entry on oeis.org

35, 41, 140, 164, 205, 221, 315, 369, 389, 391, 560, 656, 689, 775, 820, 875, 884, 1025, 1189, 1260, 1476, 1556, 1564, 1565, 1625, 1715, 1739, 1781, 1845, 1855, 1989, 2009, 2240, 2624, 2756, 2835, 3100, 3280, 3321, 3500, 3501, 3519, 3536, 3865, 3869, 4100, 4105

Offset: 1

Felix Huber, Jan 18 2025

nonn

Corresponding long legs in A380073, short legs in A380074.

Subsequence of A009000 and supersequence of A088319.

			35 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.
206125 is twice in the sequence because 31525^2 + 203700^2 = 206125^2 and 31525 + 203700 = 485^2 as well as 94588^2 + 183141^2 = 206125^2 and 94588 + 183141 = 527^2.

Felix Huber, Table of n, a(n) for n = 1..10001
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A000290, A007913, A009000, A088319, A379830, A380073, A380074.

# Calculates the first 10001 terms
A380072:=proc(M)
    local i,m,p,q,r,w,L;
    L:=[];
    m:=M^2+2*M+2;
    for p from 2 to M do
        for q to p-1 do
            if gcd(p,q)=1 and (is(p,even) or is(q,even)) then
                r:=1;
                for i in ifactors(p^2-q^2+2*p*q)[2] do
                    if is(i[2],odd) then
                        r:=r*i[1]
                    fi
                od;
                w:=r*(p^2+q^2);
                if w<=m then
                    L:=[op(L),seq(i^2*w,i=1..floor(sqrt(m/w)))]
                fi
            fi
        od
    od;
    return op(sort(L))
end proc;
A380072(4330);

A025302 Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.

Original entry on oeis.org

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 169, 173, 178, 180, 181, 193, 194, 197, 200, 202, 208, 212, 218, 225, 226, 229

Offset: 1

David W. Wilson

nonn

From Fermat's two squares theorem, every prime of the form 4k + 1 is a term (A002144). - Bernard Schott, Apr 15 2022

Donovan Johnson, Table of n, a(n) for n = 1..1000.
Art of Problem Solving, Fermat's Two Squares Theorem.
Index entries for sequences related to sums of squares.

Cf. A002144 (subsequence), A009000, A009003, A024507, A025441, A004431.

Cf. Subsequence of A001983; A004435.

a025302 n = a025302_list !! (n-1)
a025302_list = [x | x <- [1..], a025441 x == 1]

nn = 229; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, 1]] (* T. D. Noe, Apr 07 2011 *)
a[1] = 5; a[ n_] := a[n] = Module[ {s = a[n - 1], t = True, j}, While[ t, s++; Do[ If[ i^2 + (j = Floor[Sqrt[s - i^2]])^2 == s && i < j, t = False; Break], {i, Sqrt[s/2]}]]; s]; (* Michael Somos, Jan 20 2019 *)

from collections import Counter
from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(1, int(lim**.5)+2)))
  s2 = filter(lambda x: x <= lim, (sum(c) for c in combinations(s, 2)))
  s2counts = Counter(s2)
  return sorted(k for k in s2counts if k <= lim and s2counts[k] == 1)
print(aupto(229)) # Michael S. Branicky, May 10 2021

A025441(a(n)) = 1. - Reinhard Zumkeller, Dec 20 2013

A071821 Numbers whose largest prime factor is of the form 4k+1.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 37, 39, 40, 41, 45, 50, 51, 52, 53, 58, 60, 61, 65, 68, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 97, 100, 101, 102, 104, 106, 109, 111, 113, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 143, 145, 146, 148, 149

Offset: 1

Benoit Cloitre, Jun 07 2002

Subsequence of A009003. - M. F. Hasler, Feb 06 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004431, A006530, A009000, A009003, A073503, A083025.

filter:= proc(n)
  max(numtheory:-factorset(n)) mod 4 = 1
end proc:
select(filter, [$1..200]); # Robert Israel, Sep 11 2020

Select[Range[2, 150], Mod[FactorInteger[#][[-1,1]], 4] == 1 &] (* Amiram Eldar, May 04 2022 *)

for(n=2, 200, if((component(component(factor(n), 1), omega(n))-1)%4==0, print1(n, ", ")))

for( n=2,99, vecmax(factor(n)[,1])%4==1 && print1(n",")) \\ M. F. Hasler, Feb 06 2009

Numbers k such that A006530(k) == 1 (mod 4).

cp's OEIS Frontend

A024507 Numbers that are the sum of 2 distinct nonzero squares (with repetition).

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A344330 Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

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A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.

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A344332 Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

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A057100 Hypotenuses of Pythagorean triangles (ordered by the product of the sides).

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A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

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A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

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