A001481 -id:A001481 - cp's OEIS frontend

A128106 Sizes of possible gaps around a Gaussian prime: 1 and the even numbers in A001481.

1, 2, 4, 8, 10, 16, 18, 20, 26, 32, 34, 36, 40, 50, 52, 58, 64, 68, 72, 74, 80, 82, 90, 98, 100, 104, 106, 116, 122, 128, 130, 136, 144, 146, 148, 160, 162, 164, 170, 178, 180, 194, 196, 200, 202, 208, 212, 218, 226, 232, 234, 242, 244, 250, 256, 260, 272, 274

Offset: 1

T. D. Noe, Feb 15 2007

nonn

For a given Gaussian prime u, the size of its gap is the minimum of norm(u-v) as v varies over all other Gaussian primes, where norm(a+b*i)=a^2+b^2. Only the small Gaussian primes 1+i and 2+i (and their associates and reflections) have gaps of diameter 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A128107, A128108, A128109.

q=12; imax=2*q^2; lst=Select[Union[Flatten[Table[2*x^2+2*y^2, {x,0,q}, {y,0,x}]]], #<=imax&]; Join[{1},Drop[lst,1]] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

def A128106_list(max):
    R = []; s = 1; sq = 1
    for n in (0..max//2):
        if n == s:
            sq += 1;
            s = sq*sq;
        for k in range(sq):
            if is_square(n-k*k):
                R.append(2*n)
                break
    R[0] = 1
    return R
A128106_list(274) # Peter Luschny, Jun 20 2014

A070176 Let s(n) be smallest number >= n which is a sum of two squares (A001481); sequence gives s(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 1, 0, 4, 3, 2, 1, 0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 3, 2, 1, 0, 3, 2, 1, 0, 0, 1, 0, 0, 4, 3, 2, 1, 0, 2, 1, 0, 2, 1, 0, 0, 2, 1, 0, 3, 2, 1, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0, 0, 2, 1, 0, 3, 2, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 0, 1, 0, 0, 2, 1, 0

Offset: 0

N. J. A. Sloane, May 13 2002

It is an unsolved problem to determine the rate of growth of this sequence.

a(A001481(n)) = 0; a(A022544(n)) > 0. [Reinhard Zumkeller, Feb 04 2012]

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.

T. D. Noe, Table of n, a(n) for n = 0..10000

a070176 n = (head $ dropWhile (< n) a001481_list) - n
a070176_list = map a070176 [0..]
-- Reinhard Zumkeller, Feb 04 2012

sumOfTwoSquaresQ[n_] := With[{r = Ceiling[Sqrt[n]]}, Do[ Which[n == x^2 + y^2, Return[True], x == r && y == r, Return[False]], {x, 0, r}, {y, x, r}]]; a[n_] := For[s = n, True, s++, If[sumOfTwoSquaresQ[s], Return[s - n]]]; Table[a[n], {n, 0, 104}](* Jean-François Alcover, May 23 2012 *)
s2s[n_]:=Module[{i=0},While[SquaresR[2,n+i]==0,i++];i]; Array[s2s,110,0] (* Harvey P. Dale, Jun 16 2012 *)

More terms from Jason Earls, Jun 15 2002

A328803 The minimum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 4, 5, 6, 6, 5, 6, 7, 8, 8, 6, 7, 8, 9, 9, 7, 8, 10, 9, 10, 11, 8, 9, 10, 12, 11, 12, 12, 9, 10, 11, 13, 12, 13, 14, 10, 11, 12, 14, 13, 15, 14, 15, 11, 12, 13, 16, 14, 16, 15, 12, 13, 16, 14, 17, 15, 17, 16, 18, 18, 13, 14, 15, 16

Offset: 1

Peter Kagey, Oct 27 2019

			For n = 14, A001481(14) = 25 = 0^2 + 5^2 = 3^2 + 4^2, so a(14) = min{0+5, 3+4} = 5.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000161, A001481.

N:= 1000: # for terms where A001481(n)<=N
for s from 0 to isqrt(N) do
   for i from 0 to s/2 do
      t:= i^2 + (s-i)^2;
      if t > N then break fi;
      if not assigned(R[t]) then R[t]:= s fi;
od od:
A1481:= sort(map(op, [indices(R)])):
seq(R[i],i=A1481); # Robert Israel, Oct 28 2019

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
from sympy import factorint
def A328803_gen(): # generator of terms
    return map(lambda n: min((a+b for a, b in diop_DN(-1,n))), filter(lambda n:(lambda m:all(d&3!=3 or m[d]&1==0 for d in m))(factorint(n)), count(0)))
A328803_list = list(islice(A328803_gen(),30)) # Chai Wah Wu, Sep 09 2022

A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

Original entry on oeis.org

1, 4, 2, 6, 5, 5, 6, 0, 6, 3, 5, 1, 2, 5, 9, 2, 8, 7, 8, 6, 9, 6, 8, 0, 9, 3, 1, 6, 1, 5, 5, 0, 8, 1, 6, 3, 6, 1, 2, 7, 6, 6, 9, 3, 6, 3, 6, 7, 7, 0, 3, 9, 0, 2, 8, 8, 7, 9, 9, 2, 2, 3, 0, 4, 4, 1, 2, 9, 6, 0, 4, 5, 2, 8, 6, 1, 5, 1, 9, 0, 1, 9, 1, 4, 6, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an equivalent of zeta(2) (= Pi^2/6 = A013661), where Euler's zeta(2) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

Close to the value of e^(3/2)/Pi.

			1.4265560635125928786968093161550816361276693636770...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025, A175647, A243379, A334448.

Cf. A344124, A344125.

Equals Sum_{i > 0} 1/A001481(i)^2.
Equals Product_{i > 0} 1/(1-A055025(i)^-2).
Equals 1/(1-prime(1)^(-2)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-2)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-4)), where prime(n) = A000040(n).
Equals (4/3)/(A243379*A334448).
Equals zeta_{2,0} (2) * zeta_{4,1} (2) * zeta_{4,3} (4), where zeta_{4,1} (2)  = A175647 and zeta_{2,0} (s) = 2^s/(2^s - 1).

A155562 Intersection of A001481 and A002479: N = a^2 + b^2 = c^2 + 2d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 17, 18, 25, 32, 34, 36, 41, 49, 50, 64, 68, 72, 73, 81, 82, 89, 97, 98, 100, 113, 121, 128, 136, 137, 144, 146, 153, 162, 164, 169, 178, 193, 194, 196, 200, 225, 226, 233, 241, 242, 256, 257, 272, 274, 281, 288, 289, 292, 306, 313, 324, 328

Offset: 1

M. F. Hasler, Jan 24 2009

Contains A155561 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) and A001105 (twice the squares) as subsequence.
From Warut Roonguthai, Oct 13 2009: (Start)
N is also of the form x^2 - 2y^2.
N = (p^2-q^2-2*r*s)^2+(r^2-s^2-2*p*q)^2
= (p^2+q^2-r^2-s^2)^2+2*(p*r-p*s-q*r-q*s)^2
= (p^2+q^2+r^2+s^2)^2-2*(p*r+p*s+q*r-q*s)^2
for some nonnegative integers p, q, r, s. (End)
Numbers k such that in the prime factorization of k, all odd primes that occur with an odd exponent are congruent to 1 (mod 8). - Robert Israel, Jun 24 2024

Robert Israel, Table of n, a(n) for n = 1..10000
Andrew D. Ionaşcu, Intersecting semi-disks and the synergy of three quadratic forms, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 5-13.

isA155562(n,/* use optional 2nd arg to get other analogous sequences */c=[2,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155562(n) & print1(n","))

from itertools import count, islice
from sympy import factorint
def A155562_gen(): # generator of terms
    return filter(lambda n:all((p & 3 != 3 and p & 7 < 5) or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A155562_list = list(islice(A155562_gen(),30)) # Chai Wah Wu, Jun 27 2022

A229140 Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares (A001481).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 2, 0, 1, 2, 4, 3, 0, 1, 2, 4, 3, 0, 1, 4, 2, 3, 5, 0, 1, 2, 6, 3, 5, 4, 0, 1, 2, 5, 3, 4, 7, 0, 1, 2, 5, 3, 7, 4, 6, 0, 1, 2, 8, 3, 6, 4, 0, 1, 5, 2, 7, 3, 6, 4, 9, 8, 0, 1, 2, 3, 6, 9, 4, 7, 5, 0, 1, 2, 9, 3, 8, 4, 7, 5, 0

Offset: 1

Ralf Stephan, Sep 15 2013

nonn

Conjecture: the values between two zeros are always distinct from each other.

			The 6th number expressible as sum of two squares A001481(6) = 8 = 2^2 + 2^2, so a(6)=2.

Cf. A001481, A385236 (largest k), A385237, A283303, A283304.

for(n=0, 300, s=sqrtint(n); forstep(i=s, 0, -1, if(issquare(n-i*i), print1(sqrtint(n-i*i), ", "); break))); \\ shift corrected by Michel Marcus, Jul 08 2025

a(n) = 0 if A001481(n) is square.

a(n) = sqrt(A001481(n)-A385236(n)^2). - Zhuorui He, Jul 08 2025

A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

Original entry on oeis.org

1, 1, 5, 4, 5, 3, 8, 3, 3, 0, 4, 7, 6, 3, 8, 8, 9, 4, 3, 9, 2, 2, 1, 0, 6, 5, 9, 4, 5, 5, 5, 5, 1, 6, 8, 2, 9, 8, 9, 8, 7, 7, 5, 1, 9, 7, 4, 4, 8, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(3) (= Apéry's constant = A002117), where Euler's zeta(3) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

			1.1545383304763889439221065945555168298987751974487...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025.

Cf. A344123, A344125.

Equals Sum_{i > 0} 1/A001481(i)^3.
Equals Product_{i > 0} 1/(1-A055025(i)^-3).
Equals 1/(1-prime(1)^(-3)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-3)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-6)), where prime(n) = A000040(n).
Equals zeta_{2,0} (3) * zeta_{4,1} (3) * zeta_{4,3} (6), where zeta_{2,0} (s) = 2^s/(2^s - 1).

A344125 Decimal expansion of Sum_{i > 0} 1/A001481(i)^4.

Original entry on oeis.org

1, 0, 6, 8, 5, 9, 2, 1, 0, 5, 6, 5, 4, 9, 9, 0, 1, 3, 5, 2, 0, 2, 9, 4, 8, 0, 2, 0, 7, 4, 3, 2, 4, 3, 6, 1, 3, 6, 1, 3, 3, 3, 5, 9, 0, 8, 1, 0, 1, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(4) (= Pi^4/90 = A013662), where Euler's zeta(4) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

			1.0685921056549901352029480207432436136133359081017...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025.

Cf. A344123, A344124.

Equals Sum_{i > 0} 1/A001481(i)^4.
Equals Product_{i > 0} 1/(1-A055025(i)^-4).
Equals 1/(1-prime(1)^(-4)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-4)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-8)), where prime(n) = A000040(n).
Equals zeta_{2,0} (4) * zeta_{4,1} (4) * zeta_{4,3} (8), where zeta_{2,0} (s) = 2^s/(2^s - 1).

A386838 Minimum number of unit squares that a straight line drawn from (0,0) to (x,y) passes through on the square lattice where x^2 + y^2 = A001481(n). If A001481(n) can be written as a sum of two squares in two or more ways, x and y are chosen such that a(n) is the least value.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 4, 0, 4, 3, 4, 0, 5, 6, 4, 7, 0, 6, 6, 8, 6, 0, 5, 8, 8, 9, 10, 0, 8, 8, 6, 10, 11, 8, 0, 9, 10, 12, 9, 12, 7, 0, 10, 10, 13, 12, 14, 12, 12, 0, 11, 10, 8, 13, 14, 14, 0, 12, 15, 12, 16, 12, 16, 12, 9, 16, 0, 13, 14, 15, 12, 18, 16, 18, 17, 0, 14

Offset: 1

Miles Englezou, Aug 05 2025

nonn

a(n) is the minimal area of the graph formed under the requirement that the straight line (0,0) to (x,y) passes through an enclosed space on the square lattice, with the graph drawn using only vertical and horizontal edges.

Every nonnegative integer n appears in this sequence. Proof: Since 2*n^2 = n^2 + n^2 then by the first formula in the formula section n + n - gcd(n,n) = n. To prove that a(m) = n when A001481(m) = 2*n^2, we have to prove that x = n and y = n is the choice such that a(m) is minimal. Let r and s be two other numbers such that 2*n^2 = r^2 + s^2. Let r > n: consequently s < n, 1 <= gcd(r,s) <= s, and s - gcd(r,s) >= 0. If r + s - gcd(r,s) <= n, then s - gcd(r,s) < 0. But s - gcd(r,s) >= 0. Therefore r + s - gcd(r,s) >= r > n, and a(m) = n.

			a(4) = 0 since A001481(4) = 4 and 4 = 2^2 + 0^2. A straight line from (0,0) to (2,0) stays on the x axis and therefore passes through no unit squares.
a(5) = 2 since A001481(5) = 5 and 5 = 2^2 + 1^2. A straight line from (0,0) to (2,1) passes through two unit squares. It looks like this:
      _ _ (2,1)
     |_|_|
(0,0)
a(6) = 2 since A001481(6) = 8 and 8 = 2^2 + 2^2. A straight line from (0,0) to (2,2) passes through two unit squares. It looks like this:
        _ (2,2)
      _|_|
     |_|
(0,0)
a(16) = 6 since A001481(16) = 29 and 29 = 5^2 + 2^2. A straight line from (0,0) to (5,2) passes through six unit squares. It looks like this:
          _ _ _ (5,2)
      _ _|_|_|_|
     |_|_|_|
(0,0)
a(14) = 0 since A001481(14) = 25 and 25 = 5^2 + 0^2 = 4^2 + 3^2. x + y - gcd(x,y) is minimal for x = 5 and y = 0 and is equal to zero, therefore a(14) = 0.

Miles Englezou, Table of n, a(n) for n = 1..10000

Cf. A001481, A328803.

a(n) = my(f, S, T = []); (f(n) = my(P = []); for(x=0, sqrtint(n), my(y2 = n - x^2); if(issquare(y2), my(y = sqrtint(y2)); if(x <= y, P = concat(P, [[x, y]])))); return(P)); S = f(n); if(#S == 0, return(0), for(k = 1, #S, T = concat(T, S[k][1] + S[k][2] - gcd(S[k][1], S[k][2]))); return(vecmin(T))) \\ function will also return 0 for n not in A001481 so any loop of a(n) must filter n

For x and y defined in the title, a(n) = x + y - gcd(x,y).
a(n) = 0 when A001481(n) is square.
a(n) = k when A001481(n) = 2*k^2, for k >= 0.
a(n) = A328803(n) - gcd(x,y) for A001481(n) = x^2 + y^2 with exactly one decomposition into a sum of two squares.

A071011 Numbers n such that n is a sum of 2 squares (i.e., n is in A001481(k)) and sigma(n) == 0 (mod 4).

Original entry on oeis.org

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 340, 365, 370, 377, 410, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 820, 865, 884, 890, 901, 905

Offset: 1

Benoit Cloitre, May 19 2002

It is conjectured that if m is not a sum of 2 squares (i.e., m is in A022544(k)) sigma(m) == 0 (mod 4).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544.

Select[Range[10^3], And[SquaresR[2, #] > 0, Divisible[DivisorSigma[1, #], 4]] &] (* Michael De Vlieger, Jul 30 2017 *)

for(n=1,1000,if(1-sign(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))))+sigma(n)%4==0,print1(n,",")))

from math import prod
from itertools import count, islice
from sympy import factorint
def A071011_gen(): # generator of terms
    return filter(lambda n:(lambda f:all(p & 3 != 3 or e & 1 == 0 for p, e in f) and prod((p**(e+1)-1)//(p-1) & 3 for p, e in f) & 3 == 0)(factorint(n).items()),count(0))
A071011_list = list(islice(A071011_gen(),30)) # Chai Wah Wu, Jun 27 2022

cp's OEIS Frontend

A128106 Sizes of possible gaps around a Gaussian prime: 1 and the even numbers in A001481.

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A070176 Let s(n) be smallest number >= n which is a sum of two squares (A001481); sequence gives s(n) - n.

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A328803 The minimum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

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A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

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A155562 Intersection of A001481 and A002479: N = a^2 + b^2 = c^2 + 2d^2 for some integers a,b,c,d.

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PARI

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A229140 Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares (A001481).

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PARI

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A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

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A344125 Decimal expansion of Sum_{i > 0} 1/A001481(i)^4.

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A386838 Minimum number of unit squares that a straight line drawn from (0,0) to (x,y) passes through on the square lattice where x^2 + y^2 = A001481(n). If A001481(n) can be written as a sum of two squares in two or more ways, x and y are chosen such that a(n) is the least value.

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