A072437 -id:A072437 - cp's OEIS frontend

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127

Offset: 1

N. J. A. Sloane

nonn

Also numbers with no prime factors of form 4*k+1.
m is a term iff A072438(m) = m.
Density 0. - Charles R Greathouse IV, Apr 16 2012
Closed under multiplication. Primitive elements are A045326, 2 and the primes of form 4*k+3. - Jean-Christophe Hervé, Nov 17 2013

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Evan M. Bailey, Table of n, a(n) for n = 1..20000 (Terms 1..1000 from T. D. Noe)
Evan M. Bailey, a004144.cpp.
Steven R. Finch, Landau-Ramanujan Constant [Broken link]
Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
Daniel Shanks, Non-hypotenuse numbers, Fib. Quart., Vol. 13, No. 4 (1975), pp. 319-321.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for sequences related to sums of squares.

Complement of A009003.
Cf. A000290, A002145, A005089, A072437, A244659, A244662.
The subsequence of primes is A045326.

import Data.List (elemIndices)
a004144 n = a004144_list !! (n-1)
a004144_list = map (+ 1) $ elemIndices 0 a005089_list
-- Reinhard Zumkeller, Jan 07 2013

fQ[n_] := If[n > 1, First@ Union@ Mod[ First@# & /@ FactorInteger@ n, 4] != 1, True]; Select[ Range@ 127, fQ]
A004144 = Select[Range[127],Length@Reduce[s^2 + t^2 == s # && s > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 09 2020 *)

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

list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=1,lim, if(u[i]==0, listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

A005089(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013

The number of terms below x is ~ (A * x / sqrt(log(x))) * (1 + C/log(x) + O(1/log(x)^2)), where A = A244659 and C = A244662 (Shanks, 1975). - Amiram Eldar, Jan 29 2022

More terms from Reinhard Zumkeller, Jun 17 2002

Name clarified by Evan M. Bailey, Sep 17 2019

A001842 Expansion of Sum_{n>=0} x^(4n+3)/(1 - x^(4n+3)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 1, 2, 0, 0, 2, 1, 1, 2, 1, 1, 1, 1, 0, 2, 0, 0, 2, 2, 1, 2, 0, 1, 2, 0, 1, 3, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 0, 3, 1, 2, 2, 1, 0, 2, 0, 1, 2, 0, 1

Offset: 0

N. J. A. Sloane

Number of divisors of n of the form 4*k+3. - Reinhard Zumkeller, Apr 18 2006

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 244.

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001227, A001620, A001826, A072437, A256846.

with(numtheory): seq(add(binomial(d,3) mod 2, d in divisors(n)), n=0..100); # Ridouane Oudra, Nov 19 2019

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 4] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 4] == 3 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

a(n) = if(n<1, 0, sumdiv(n, d, d%4 == 3)); \\ Amiram Eldar, Nov 25 2023

a(A072437(n)) = 0. - Benoit Cloitre, Apr 24 2003
a(n) = A001227(n) - A001826(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 11 2019
a(n) = Sum_{d|n} (binomial(d,3) mod 2). - Ridouane Oudra, Nov 19 2019
Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (1 - gamma)/4 = A256846 - (1 - A001620)/4 = -0.180804... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A097706 Part of n composed of prime factors of form 4k+3.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, 3, 1, 7, 3, 1, 1, 9, 19, 1, 21, 11, 23, 3, 1, 1, 27, 7, 1, 3, 31, 1, 33, 1, 7, 9, 1, 19, 3, 1, 1, 21, 43, 11, 9, 23, 47, 3, 49, 1, 3, 1, 1, 27, 11, 7, 57, 1, 59, 3, 1, 31, 63, 1, 1, 33, 67, 1, 69, 7, 71, 9, 1, 1, 3, 19, 77, 3, 79, 1, 81, 1, 83, 21

Offset: 1

Ralf Stephan, Aug 30 2004

Largest term of A004614 that divides n. - Peter Munn, Apr 15 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Equivalent sequence for distinct prime factors: A170819.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A170818 (4k+1), A072436 (not 4k+3), A248909 (6k+1), A343431 (6k+5).
Range of values: A004614.
Positions of 1's: A072437.
Cf. also A065338, A065339, A260728, A286363.

a:= n-> mul(`if`(irem(i[1], 4)=3, i[1]^i[2], 1), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}]; Array[a, 100] (* Jean-François Alcover, Jun 16 2015, updated May 29 2019 *)

a(n)=local(f); f=factor(n); prod(k=1, matsize(f)[1], if(f[k, 1]%4<>3, 1, f[k, 1]^f[k, 2]))

from sympy import factorint
from operator import mul
def a072436(n):
    f=factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])
def a(n): return n/a072436(n) # Indranil Ghosh, May 08 2017

from math import prod
from sympy import factorint
def A097706(n): return prod(p**e for p, e in factorint(n).items() if p & 3 == 3) # Chai Wah Wu, Jun 28 2022

a(n) = n/A072436(n).
a(A004614(n)) = A004614(n).
a(A072437(n)) = 1.
a(n) = A000265(n)/A170818(n). - Peter Munn, Apr 15 2021

A005091 Number of distinct primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 0, 0, 1

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Peter Koymans, On Dirichlet biquadratic fields, arXiv:2001.05350 [math.NT], 2020.

Cf. A001221, A005089, A005094.

Cf. A079261, A027748, A072437, A187811.

a005091 = sum . map a079261 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013

[0] cat [#[p:p in PrimeDivisors(n)| p mod 4 eq 3]: n in [2..100]]; // Marius A. Burtea, Nov 19 2019

[0] cat [&+[Binomial(p,3) mod 2:p in PrimeDivisors(n)]:n in [2..100]]; // Marius A. Burtea, Nov 19 2019

with(numtheory): seq(add(binomial(p,3) mod 2, p in factorset(n)), n=1..100); # Ridouane Oudra, Nov 19 2019

f[n_]:=Length@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==3&]; Table[f[n],{n,102}] (* Ray Chandler, Dec 18 2011 *)

for(n=1,100,print1(sumdiv(n,d,isprime(d)*if((d-3)%4,0,1)),","))

from sympy import primefactors
def A005091(n): return sum(1 for p in primefactors(n) if p&3==3) # Chai Wah Wu, Jul 07 2024

Additive with a(p^e) = 1 if p = 3 (mod 4), 0 otherwise.
From Reinhard Zumkeller, Jan 07 2013: (Start)
a(n) = Sum_{k=1..A001221(n)} A079261(A027748(n,k)).
a(A072437(n)) = 0.
a(A187811(n)) > 0. (End)
a(n) = Sum_{p|n} (binomial(p,3) mod 2), where p is a prime. - Ridouane Oudra, Nov 19 2019

A050466 a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

0, 0, 1, 0, 0, 8, 1, 0, 27, 0, 1, 64, 0, 8, 126, 0, 0, 216, 1, 0, 370, 8, 1, 512, 0, 0, 730, 64, 0, 1008, 1, 0, 1358, 0, 126, 1728, 0, 8, 2198, 0, 0, 2960, 1, 64, 3402, 8, 1, 4096, 343, 0, 4914, 0, 0, 5840, 126, 512, 6886, 0, 1, 8064, 0, 8, 9991, 0

Offset: 1

N. J. A. Sloane, Dec 23 1999

From Robert G. Wilson v, Mar 26 2015: (Start)
a(n) = 0 for n = 1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, ... (A072437).
a(n) = 1 for n = 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ... (A002145). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A007331, A050462, A050471, A175572.
Cf. A050464, A050465, A050467.
Cf. A002145, A072437.

a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 3 &])^3]; Array[a, 64] (* Robert G. Wilson v, Mar 26 2015 *)
a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, ((n/d % 4)== 3)* d^3); \\ Michel Marcus, Mar 26 2015

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A007331(n) - A050462(n).
a(n) = A050462(n) - A050471(n).
a(n) = (A007331(n) - A050471(n))/2.
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Pi^4/192 - A175572/2 = 0.0128667399315... . (End)

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015

A072436 Remove prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 13, 2, 5, 16, 17, 2, 1, 20, 1, 2, 1, 8, 25, 26, 1, 4, 29, 10, 1, 32, 1, 34, 5, 4, 37, 2, 13, 40, 41, 2, 1, 4, 5, 2, 1, 16, 1, 50, 17, 52, 53, 2, 5, 8, 1, 58, 1, 20, 61, 2, 1, 64, 65, 2, 1, 68, 1, 10, 1, 8, 73, 74, 25, 4, 1, 26, 1, 80, 1, 82, 1, 4, 85, 2, 29

Offset: 1

Reinhard Zumkeller, Jun 17 2002

a(n) <= n; a(a(n)) = a(n); for all factors p^m of a(n): p=2 or p=4*k+1.

			a(90) = a(2*3*3*5) = a(2*(4*0+3)^2*(4*1+1)^1) = 2*1^2*5 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072438, A072437, A002144, A002145, A065338.

Equals n / A097706(n).

a:= n-> mul(`if`(irem(i[1], 4)=3, 1, i[1]^i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := n/Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, if ((f[k,1] % 4) == 3, f[k,1]=1)); factorback(f); \\ Michel Marcus, May 08 2017

from sympy import factorint
from operator import mul
def a(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])# Indranil Ghosh, May 08 2017

Multiplicative with a(p) = (if p==3 (mod 4) then 1 else p).

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 *)

A187811 Numbers having at least one prime factor of form 4*k+3.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 14, 15, 18, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 72, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99

Offset: 1

Reinhard Zumkeller, Jan 07 2013

nonn

A005091(a(n)) > 0. - Reinhard Zumkeller, Jan 07 2013

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

Cf. A072437 (complement); A002145, A009003.

import Data.List (findIndices)
a187811 n = a187811_list !! (n-1)
a187811_list = map (+ 1) $ findIndices (> 0) a005091_list
-- Reinhard Zumkeller, Jan 07 2013

pfQ[n_]:=AnyTrue[Transpose[FactorInteger[n]][[1]],Mod[#,4]==3&]; Select[ Range[100],pfQ] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2016 *)

is(n)=if(n%4==3,return(1)); my(f=factor(n)[,1]%4); for(i=1,#f, if(f[i]==3, return(1))); 0 \\ Charles R Greathouse IV, Sep 01 2015

a(n) ~ n. - Charles R Greathouse IV, Sep 01 2015

A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 4).

Original entry on oeis.org

1, 1, 4, 2, 4, 4, 8, 4, 12, 4, 12, 8, 12, 8, 16, 8, 16, 12, 20, 8, 32, 12, 24, 16, 20, 12, 36, 16, 28, 16, 32, 16, 48, 16, 32, 24, 36, 20, 48, 16, 40, 32, 44, 24, 48, 24, 48, 32, 56, 20, 64, 24, 52, 36, 48, 32, 80, 28, 60, 32, 60, 32, 96, 32, 48, 48, 68, 32

Offset: 1

José María Grau Ribas, Jan 17 2012

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A072437.
Cf. A175647, A243381.
Cf. A079458, A062570.

with(numtheory):
a := n->add(jacobi(-1,d)*mobius(d)*n/d, d in divisors(n)):
seq(a(n), n = 1..60); # Peter Bala, Dec 26 2023

ar[p_,s_] := Which[Mod[p,4]==1, p^(s-1)*(p-1), Mod[p,4]==3, p^(s-1)*(p+1), True,p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]

A204617(n) = { my(f=factor(n),p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p,1,if(1==(p%4),p-1,p+1))); }; \\ Antti Karttunen, Nov 16 2021

a(n) = phi(n) if n is in A072437.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - Amiram Eldar, Dec 24 2022
a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - Peter Bala, Dec 26 2023
a(n) = A079458(n)/A062570(n). - Ridouane Oudra, Jun 04 2024

A285510 Numbers k such that the average of the squarefree divisors of k is an integer.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101

Offset: 1

Ilya Gutkovskiy, Apr 20 2017

nonn

Numbers n such that A034444(n)|A048250(n).
Numbers n such that 2^omega(n)|psi(rad(n)), where omega() is the number of distinct prime divisors (A001221), psi() is the Dedekind psi function (A001615) and rad() is the squarefree kernel (A007947).
From Robert Israel, Apr 24 2017: (Start)
All odd numbers are in the sequence.
A positive even number is in the sequence if and only if at least one of its prime factors is in A002145.
Thus this is the complement of 2*A072437 in the positive numbers.
(End)

			44 is in the sequence because 44 has 6 divisors {1, 2, 4, 11, 22, 44} among which 4 are squarefree {1, 2, 11, 22} and (1 + 2 + 11 + 22)/4 = 9 is an integer.

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

Cf. A001221, A001615, A002145, A003601, A005117, A007947, A023886, A034444, A048250, A072437, A078174, A103826, A206778.

filter:= n -> n::odd or has(numtheory:-factorset(n) mod 4, 3):
select(filter, [$1..1000]); # Robert Israel, Apr 24 2017

Select[Range[100], IntegerQ[Total[Select[Divisors[#], SquareFreeQ]] / 2^PrimeNu[#]] &]
Select[Range[110],IntegerQ[Mean[Select[Divisors[#],SquareFreeQ]]]&] (* Harvey P. Dale, Apr 11 2018 *)
Select[Range[100], IntegerQ[Times @@ ((1 + FactorInteger[#][[;; , 1]])/2)] &] (* Amiram Eldar, Jul 01 2022 *)

a(n) ~ n (conjecture).

Conjecture is true, since A072437 has density 0. - Robert Israel, Apr 24 2017

cp's OEIS Frontend

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

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A001842 Expansion of Sum_{n>=0} x^(4*n+3)/(1 - x^(4*n+3)).

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A097706 Part of n composed of prime factors of form 4k+3.

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A005091 Number of distinct primes = 3 mod 4 dividing n.

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A050466 a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

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A072436 Remove prime factors of form 4*k+3.

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A001842 Expansion of Sum_{n>=0} x^(4n+3)/(1 - x^(4n+3)).