A004613 -id:A004613 - cp's OEIS frontend

A062327 Number of divisors of n over the Gaussian integers.

1, 3, 2, 5, 4, 6, 2, 7, 3, 12, 2, 10, 4, 6, 8, 9, 4, 9, 2, 20, 4, 6, 2, 14, 9, 12, 4, 10, 4, 24, 2, 11, 4, 12, 8, 15, 4, 6, 8, 28, 4, 12, 2, 10, 12, 6, 2, 18, 3, 27, 8, 20, 4, 12, 8, 14, 4, 12, 2, 40, 4, 6, 6, 13, 16, 12, 2, 20, 4, 24, 2, 21, 4, 12, 18, 10, 4, 24, 2, 36, 5, 12, 2, 20, 16, 6

Offset: 1

Reiner Martin, Jul 12 2001

Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e., one of 1, i, -1, -i).
a(A004614(n)) = A000005(n). - Vladeta Jovovic, Jan 23 2003
a(A004613(n)) = A000005(n)^2. - Benedikt Otten, May 22 2013

			For example, 5 has divisors 1, 1+2i, 2+i and 5.

Cf. A027748, A124010.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319442.

a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e                  = 2 * e + 1
   f p e | p `mod` 4 == 1 = (e + 1) ^ 2
         | otherwise      = e + 1
-- Reinhard Zumkeller, Oct 18 2011

a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
                 i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2021

Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* Alonso del Arte, Jan 25 2011 *)
DivisorSigma[0,Range[90],GaussianIntegers->True] (* Harvey P. Dale, Mar 19 2017 *)

a(n)=
{
    my(r=1,f=factor(n));
    for(j=1,#f[,1], my(p=f[j,1],e=f[j,2]);
        if(p==2,r*=(2*e+1));
        if(p%4==1,r*=(e+1)^2);
        if(p%4==3,r*=(e+1));
    );
    return(r);
}  \\ Joerg Arndt, Dec 09 2016

Presumably a(n) = 2 iff n is a rational prime == 3 mod 4 (see A045326). - N. J. A. Sloane, Jan 07 2003, Feb 23 2007

Multiplicative with a(2^e) = 2*e+1, a(p^e) = e+1 if p mod 4=3 and a(p^e) = (e+1)^2 if p mod 4=1. - Vladeta Jovovic, Jan 23 2003

A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 3, 8, 9, 2, 3, 12, 1, 6, 3, 16, 1, 18, 3, 4, 9, 6, 3, 24, 1, 2, 27, 12, 1, 6, 3, 32, 9, 2, 3, 36, 1, 6, 3, 8, 1, 18, 3, 12, 9, 6, 3, 48, 9, 2, 3, 4, 1, 54, 3, 24, 9, 2, 3, 12, 1, 6, 27, 64, 1, 18, 3, 4, 9, 6, 3, 72, 1, 2, 3, 12, 9, 6, 3, 16, 81, 2, 3, 36, 1, 6, 3, 24, 1, 18, 3

Offset: 1

Reinhard Zumkeller, Oct 29 2001

			a(120) = a(2*2*2*3*5) = a(2)*a(2)*a(2)*a(3)*a(5) = 2*2*2*3*1 = 24.
a(150) = a(2*3*5*5) = a(2)*a(3)*a(5)*a(5) = 2*3*1*1 = 6.
a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = 2*3*1*3 = 18.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A039702, A000040, A003586, A007814, A065339, A065331.

a065338 1 = 1
a065338 n = (spf `mod` 4) * a065338 (n `div` spf) where spf = a020639 n
-- Reinhard Zumkeller, Nov 18 2011

a[1] = 1; a[n_] := a[n] = Mod[p = FactorInteger[n][[1, 1]], 4]*a[n/p]; Table[ a[n], {n, 1, 100} ] (* Jean-François Alcover, Jan 20 2012 *)

a(n)=my(f=factor(n)); prod(i=1,#f~, (f[i,1]%4)^f[i,2]) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = 1 if n = 1, otherwise (A020639(n) mod 4) * n / A020639(n).
a(n) = (2^A007814(n)) * (3^A065339(n)).
a(n) <= n.
a(a(n)) = a(n).
a(x) = x iff x = 2^i * 3^j for i, j >= 0.
a(A003586(n)) = A003586(n).
a(A065331(n)) = A065331(n).
a(A004613(n)) = 1; A065333(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Dirichlet g.f.: (1/(1-2^(-s+1))) * Product_{prime p=4k+1} (1/(1-p^(-s))) * Product_{prime p=4k+3} 1/(1-3*p^(-s)). - Ralf Stephan, Mar 28 2015

A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

Original entry on oeis.org

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 27625, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125

Offset: 1

N. J. A. Sloane

nonn

a(n) is least term of A054994 with exactly n divisors. - Ray Chandler, Jan 05 2012
From Jianing Song, Apr 24 2021: (Start)
a(n) is the smallest k such that A004018(k) = 4n.
Also a(n) is the smallest index of n in A002654.
a(n) is the smallest term in A004613 that has exactly n divisors.
This is a subsequence of A054994. (End)

			4225 = 5^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 4 with exactly 9 divisors, so a(9) = 4225. - _Jianing Song_, Apr 24 2021

Ray Chandler, Table of n, a(n) for n = 1..1432 (a(1433) exceeds 1000 digits).

Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.

(* This program is not convenient to compute huge terms - A054994 is assumed to be computed with maxTerm = 10^16 *) a[n_] := Catch[ For[k = 1, k <= Length[A054994], k++, If[DivisorSigma[0, A054994[[k]]] == n, Throw[A054994[[k]]]]]]; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jan 21 2013, after Ray Chandler *)

primelist(d,r,l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p,d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A018782(n)=my(pf=prodR(n, n), a=1, b, v=primelist(4, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bJianing Song, Apr 25 2021, following R. J. Mathar's program for A005179.

A000446(n) = min(a(2n-1), a(2n)) for n > 1.
A124980(n) = min(a(2n-1), a(2n)).
A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).
A093195(n) = min(a(2n), a(2n+1)).
From Jianing Song, Apr 24 2021: (Start)
If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 4. The smallest n such that the equality does not hold is n = 16.
a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)

A072437 Numbers with no prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, 26, 29, 32, 34, 37, 40, 41, 50, 52, 53, 58, 61, 64, 65, 68, 73, 74, 80, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 169, 170, 173, 178, 181, 185, 193, 194

Offset: 1

Reinhard Zumkeller, Jun 17 2002

nonn

m is a term iff A072436(m) = m.

These numbers have density zero (Pollack).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 119. alternate link

Cf. A004144, A002144, A002145, A004613 (odd terms).
A097706(a(n)) = 1.
Cf. A187811 (complement).

import Data.List (elemIndices)
a072437 n = a072437_list !! (n-1)
a072437_list = map (+ 1) $ elemIndices 0 a005091_list
-- Reinhard Zumkeller, Jan 07 2013

npfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(Mod[#,4]==3&)]==0; Select[Range[200],npfQ] (* _Harvey P. Dale, Nov 12 2013 *)

is(n)=n==1||vecmax(factor(n)[,1]%4)<3 \\ Charles R Greathouse IV, Apr 16 2012

n>0 such that A001842(n)=0. - Benoit Cloitre, Apr 24 2003
A005091(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013
A065339(a(n)) = 0 . - R. J. Mathar, Jan 28 2025

A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

Original entry on oeis.org

1, 3, 7, 11, 19, 21, 23, 31, 33, 43, 47, 57, 59, 67, 69, 71, 77, 79, 83, 93, 103, 107, 127, 129, 131, 133, 139, 141, 151, 161, 163, 167, 177, 179, 191, 199, 201, 209, 211, 213, 217, 223, 227, 231, 237, 239, 249, 251, 253, 263, 271, 283, 301, 307, 309, 311, 321, 329

Offset: 1

Arnaud Vernier, Oct 29 2009

Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by Gabriel Conant, Apr 18 2016

If a term divides the sum of two squares, then it divides each of the two numbers individually. Moreover, only the numbers in this sequence have this property. See link for proof. - V Sai Prabhav, Jul 15 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
V Sai Prabhav, Proof for the comment

Cf. A002145, A004613, A004614, A005117, A231754.

Cf. A175647, A243379.

N:= 1000: # to get all terms <= N
S:= {1};
for p from 3 by 4 to N do
  if isprime(p) then
    S:= S union select(`<=`, map(t -> t*p, S),N)
  fi
od:
sort(convert(S,list)); # Robert Israel, Apr 18 2016

Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* Giovanni Resta, Apr 18 2016 *)

isok(n) = if (! issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 3, return (0))); 1 \\ Michel Marcus, Sep 04 2013

A005117 INTERSECT A004614. - R. J. Mathar, Nov 05 2009

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A243379/(2*sqrt(A175647)) = 0.4165140462... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Edited by Zak Seidov, Oct 30 2009

Narrowed definition down to squarefree numbers - R. J. Mathar, Nov 05 2009

A246272 Starting from n, the number of iterations of A003961 needed before the result has only prime factors of the form 4k+1 (a(1) = 0). [Where A003961(n) shifts the prime factorization of n one step towards larger primes].

Original entry on oeis.org

0, 2, 1, 2, 0, 5, 2, 2, 1, 9, 1, 5, 0, 2, 4, 2, 0, 5, 2, 9, 8, 2, 1, 5, 0, 6, 1, 2, 0, 23, 1, 2, 1, 5, 3, 5, 0, 2, 1, 9, 0, 49, 2, 2, 4, 9, 1, 5, 2, 9, 5, 6, 0, 5, 7, 2, 4, 2, 1, 23, 0, 2, 8, 2, 0, 5, 2, 5, 1, 9, 1, 5, 0, 6, 4, 2, 2, 23, 2, 9, 1, 5, 1, 49, 0, 2, 8, 2, 0, 23, 6, 9, 1, 6, 4, 5, 0, 2, 1, 9

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Among the first 10000 terms, of which 4371 are primes, there are 92 distinct values in total (of which 23 are primes), the most common of them being: 1600 x 2, 1324 x 5, 1131 x 1, 1074 x 0, 571 x 4, 557 x 6, 538 x 9, 409 x 23, 404 x 3, 378 x 11, 211 x 8, 197 x 15, 131 x 12, 130 x 24, 128 x 49, 119 x 10, 95 x 7, 95 x 76, 92 x 22, 80 x 32, 70 x 14, 53 x 20, 47 x 77, 44 x 28, 29 x 17, 27 x 58, 24 x 21, 24 x 64, 23 x 13, 22 x 31, 22 x 39, 20 x 25, 19 x 48.
In contrast to A246271, here it holds that a(u) <= a(u*v) >= a(v) for any u, v. This follows because a number n has reached the "state of 4k+1 purity" (meaning that A065338(n) = 1) only if all its possible divisors have reached it as well.
This explains why for example 23 seems to be so common value. The reason is, that it occurs for the first time as a(30), and 30, being the third primorial is a product of three commonest primes: 30 = 2*3*5, thus for any n which is a multiple of 30, a(n) >= 23. Similarly a(42) = 49, a(84) = 49, a(126) = 49, so any number k which has 2, 3 and 7 as its prime factors must have a(k) >= 49.

			Consider n = 6 = 2*3 = p_1 * p_2. Five is the least number of iterations of A003961(n) (which increments by one the prime indices of prime factorization of n), before both primes are of the form 4k+1:
  p_2 = 3, p_3 = 5 (4k+3 & 4k+1),
  p_3 = 5, p_4 = 7 (4k+1 & 4k+3),
  p_4 = 7, p_5 = 11 (4k+3 & 4k+3),
  p_5 = 11, p_6 = 13 (4k+3 & 4k+1),
  p_6 = 13, p_7 = 17 (4k+1 & 4k+1),
thus a(6) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10001

A004613 gives the positions of zeros.
A246349 gives the positions of records and A246350 the corresponding values.
Cf. A003961, A065338, A246269, A246270, A246271, A246277.

default(primelimit, 2^22)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A065338(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f);
A246272(n) = {my(i); i=0; while((A065338(n)!=1), i++; n = A003961(n)); i};
for(n=1, 10001, write("b246272.txt", n, " ", A246272(n)));

(define (A246272 n) (let loop ((i 0) (n n)) (if (= 1 (A065338 n)) i (loop (+ i 1) (A003961 n)))))

;; Requires memoizing definec-macro.
(definec (A246272 n) (if (= 1 (A065338 n)) 0 (+ 1 (A246272 (A003961 n)))))

If A065338(n) = 1, a(n) = 0, otherwise 1 + a(A003961(n)).
Other identities:
a(n) = a(A007947(n)) for all n. [Duplicate prime factors have no effect on the result].

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

A202057 Numbers which are not perfect squares and such that all prime divisors are congruent to 1 or 2 mod 4.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 20, 26, 29, 32, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 97, 101, 104, 106, 109, 113, 116, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 170, 173, 178, 181, 185, 193, 194, 197, 200, 202, 205, 208, 212, 218, 221, 226, 229, 232, 233, 241, 244, 250, 257, 260, 265, 269, 272, 274

Offset: 1

Artur Jasinski, Dec 10 2011

nonn

This sequence follows conjecture from A201278 that Mordell's elliptic curve x^3-y^2 = d can contain points {x,y} with quadratic extension sqrt(k) over rationals if and only k belongs to this sequence.

Members of A072437 that are not perfect squares. - Franklin T. Adams-Watters, Dec 15 2011

			a(3)=8 because 8 isn't perfect square and only one prime divisor 2 is congruent to 2 mod 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004613, A201278, A072437.

aa = {}; Do[pp = FactorInteger[j]; if = False; Do[If[Mod[pp[[n]][[1]], 4] == 3 || Mod[pp[[n]][[1]], 4] == 0, if = True], {n, 1, Length[pp]}]; If[if == False, If[IntegerQ[Sqrt[j]] == False, AppendTo[aa, j]]], {j, 2, 200}]; aa
seqQ[n_] := !IntegerQ@Sqrt[n] && AllTrue[FactorInteger[n][[;; , 1]], MemberQ[{1, 2}, Mod[#, 4]] &]; Select[Range[300], seqQ] (* Amiram Eldar, Mar 21 2020 *)

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... .

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

cp's OEIS Frontend

A062327 Number of divisors of n over the Gaussian integers.

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A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.

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Haskell

Mathematica

PARI

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A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

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Mathematica

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A072437 Numbers with no prime factors of form 4*k+3.

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Haskell

Mathematica

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A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

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Maple

Mathematica

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A246272 Starting from n, the number of iterations of A003961 needed before the result has only prime factors of the form 4k+1 (a(1) = 0). [Where A003961(n) shifts the prime factorization of n one step towards larger primes].

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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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