A004215 -id:A004215 - cp's OEIS frontend

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

1, -4, 2, 8, -4, -8, -8, 16, 6, -12, 8, 8, -8, -24, 0, 16, 12, -16, 10, 24, -8, -16, -24, 16, 8, -28, 8, 32, -16, -8, 0, 32, 6, -32, 16, 16, -12, -40, -24, 16, 24, -16, 16, 40, -8, -40, 0, 32, 24, -36, 10, 16, -24, -24, -32, 48, 0, -32, 24, 24, -16, -40, 0, 48, 12, -16, 16, 56, -16, -32, -48, 16, 30, -64, 8, 40, -24

Offset: 0

Michael Somos, May 04 2003

sign

Euler transform of period 4 sequence [ -4,-4,-4,-3,...].

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

A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.
Cf. A274327, A285895.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 07 2017

CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* Jean-François Alcover, Sep 19 2016 *)

a(n)=if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4),n))

G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A125084 Cubes which have a partition as the sum of 3 squares.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 512, 729, 1000, 1331, 1728, 2197, 2744, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 64000, 68921, 74088, 79507, 85184, 91125

Offset: 1

Artur Jasinski, Nov 20 2006, Nov 21 2006, Nov 22 2006

nonn

If n is of the form 4^i*(8j+7) (where i>=0, j>=0) then n^3 is not in the sequence because n^3 is of the mentioned form so n^3 is in A004215 hence according to the definition n^3 is not in this sequence (see formula for A004215). Hence 7^3, 15^3, 23^3, 28^3, 31^3, 39^3, ... are not in the sequence. Is there a number n such that n^3 is not in the sequence but n is not of the form 4^i*(8j+7)? - Farideh Firoozbakht, Nov 23 2006
A number n^3 belongs to this sequence if and only if n is sum of three squares. Proof is immediate from Catalan's identity (x^2 + y^2 + z^2)^3 = x^2*(3*z^2 - x^2 - y^2)^2 + y^2*(3*z^2 - x^2 - y^2)^2 + z^2*(z^2 - 3*x^2 - 3*y^2)^2. - Artur Jasinski, Dec 09 2006
If n = a^2 + b^2 + c^2, then n^3 = (n*a)^2 + (n*b)^2 + (n*c)^2. Conversely, suppose there were an n such that n^3 is in A000378 but n is not. Then n must be of form 4^k*(8i+7). But n^3 would also be of the form 4^k*(8i+7) and thus n^3 would not be in A000378, contradicting the original assumption. This argument is easily extended to all odd powers, i.e., n^(2k+1) is in A000378 iff n is in A000378. - Ray Chandler, Feb 03 2009

			125 is in the sequence because
  125 = 5^3 = 0^2 + 2^2 + 11^2
            = 0^2 + 5^2 + 10^2
            = 3^2 + 4^2 + 10^2
            = 5^2 + 6^2 +  8^2.
   27 = 3^3 = 1^2 + 1^2 +  5^2, so  27 is a term.
  125 = 5^3 = 0^2 + 2^2 + 11^2, so 125 is a term.
  216 = 6^3 = 2^2 + 4^2 + 14^2, so 216 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A000378, A000578, A004215.

Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* Ray Chandler, Nov 23 2006 *)

isA125084(n)={ local(cnt,a,b) ; cnt=0 ; a=0; while(a^2<=n, b=0 ; while(b<=a && a^2+b^2<=n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ; } { for(n=1,300, if(isA125084(n^3), print1(n^3,", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 23 2006

a(n) = A000378(n)^3.

Equals A000578 INTERSECT A000378.

Corrected and extended by Farideh Firoozbakht, Ray Chandler and R. J. Mathar, Nov 23 2006

A278494 Primes p for which there does not exist any such integer k that k - A002828(k) = p.

Original entry on oeis.org

2, 5, 7, 13, 17, 23, 29, 31, 37, 47, 61, 79, 89, 97, 101, 103, 109, 113, 127, 157, 167, 193, 197, 199, 223, 229, 241, 257, 269, 271, 281, 293, 313, 317, 337, 353, 359, 383, 389, 397, 401, 409, 421, 433, 439, 449, 461, 463, 487, 509, 541, 569, 577, 593, 601, 607, 631, 647, 653, 673, 677, 709, 719, 727, 751, 761, 769, 773, 797

Offset: 1

Antti Karttunen, Nov 25 2016, with additional comments Nov 28 2016

nonn

Primes that are leaves in the tree defined by edge relation parent = A255131(child), "the least squares beanstalk".
Primes p such that (A002828(1+p) <> 1), (A002828(2+p) <> 2), (A002828(3+p) <> 3) and (A002828(4+p) <> 4).
See comments in A278495 which gives the count of these primes in each range [n^2, (n+1)^2].
This is a subsequence of A045352 as no prime of the form 8n+3 ever occurs in this sequence. This stems from a more general fact that A278490 contains no numbers of the form 8n+3, because A002828(8n+7) = 4 for all n. (See A004215.)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A278490.
No common terms with A277888, some common terms with A278487.
Subsequence of A045352.
Cf. A002828, A004215, A072401, A255131, A278495, A278496.
Cf. also A263091.

A319281 Numbers of the form 16^i(16j + 1).

Original entry on oeis.org

1, 16, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 256, 257, 272, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 528, 529, 545, 561, 577, 593, 609, 625, 641, 657, 673, 689, 705, 721, 737, 753, 769, 784

Offset: 1

Jianing Song, Sep 16 2018

nonn

{a(n)} gives all positive fourth powers modulo all powers of 2, that is, positive fourth powers over 2-adic integers. So this sequence is closed under multiplication.

Jianing Song, Table of n, a(n) for n = 1..10002 (all terms <= 150000)

A158057 is a proper subsequence.
Perfect powers over 2-adic integers:
Squares: positive: A234000; negative: A004215 (negated);
Cubes: A191257;
Fourth powers: positive: this sequence; negative: A319282 (negated).

isA319281(n)= n\16^valuation(n, 16)%16==1

def A319281(n):
    if n<3: return 15*n-14
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum((((x>>(i<<2))-1)>>4)+1 for i in range(x.bit_length()>>2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 17 2025

a(n) = 15*n + O(log(n)).

A343631 X-coordinate of the points following the 3D spiral defined in A343630.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 28 2021

sign

See the main entry A343630 for details about this 3D generalization of an Ulam type spiral using the Euclidean norm.
Sequences A343632 and A343633 give the y and z coordinates.
The sequence can be seen as a table with row lengths 3*A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.
Sequence A343641 is the analog for the square spiral variant A343640.

Cf. A343632, A343633 (list of y and z-coordinates).
Cf. A343641 (variant using the sup norm => square spiral).
Cf. A342561 (variant which scans each sphere by increasing z).
Cf. A005875 (number of points on a shell with given radius).
Cf. A004215 (numbers that can't be written as sum of 3 squares => empty shells).

d=1; A343631_vec=concat([[P[1] | P<-S=A343630_row(n,d)]+(#S&&!d*=-1) | n<-[0..8]]) \\ the variable d is necessary to correct the z-scan direction in rows between A004215(2k-1) and A004215(2k).

A343632 Y-coordinate of the points following the 3D spiral defined in A343630.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 28 2021

sign

See the main entry A343630 for details about this 3D generalization of an Ulam type spiral using the Euclidean norm.
Sequences A343631 and A343633 give the x and z coordinates.
The sequence can be seen as a table with row lengths A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.
Sequence A343642 is the analog for the square spiral variant A343640.

Cf. A343631, A343633 (list of x and z-coordinates).
Cf. A343642 (variant using the sup norm => square spiral).
Cf. A342562 (variant which scans each sphere by increasing z).
Cf. A005875 (number of points on a shell with given radius).
Cf. A004215 (numbers that can't be written as sum of 3 squares => empty shells).

d=1; A343632_vec=concat([[P[2] | P<-S=A343630_row(n,d)]+(#S&&!d*=-1) | n<-[0..9]]) \\ the variable d is necessary to correct the z-scan direction in rows between A004215(2k-1) and A004215(2k).

A004014 Norms of vectors in the b.c.c. lattice.

Original entry on oeis.org

0, 3, 4, 8, 11, 12, 16, 19, 20, 24, 27, 32, 35, 36, 40, 43, 44, 48, 51, 52, 56, 59, 64, 67, 68, 72, 75, 76, 80, 83, 84, 88, 91, 96, 99, 100, 104, 107, 108, 115, 116, 120, 123, 128, 131, 132, 136, 139, 140, 144, 147, 148, 152, 155, 160, 163, 164, 168

Offset: 0

N. J. A. Sloane

Integers such that A004013(n) is nonzero. - Michael Somos, Jul 28 2014
A subsequence of A047458. The complement seems to be 4*A004215. - Andrey Zabolotskiy, Nov 11 2021
From Mohammed Yaseen, Nov 06 2022: (Start)
These are numbers of the form x^2+y^2+z^2 where x, y and z are either all even (including zero) or all odd.
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in an f.c.c. lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A000378 for simple cubic lattice. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116. (Chapter 4 section 6.7)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to b.c.c. lattice

Cf. A004013, A047458, A004215.

Union of A034045 and A017101. - Mohammed Yaseen, Nov 06 2022

f:= JacobiTheta2(0,z^4)^3+JacobiTheta3(0,z^4)^3:
S:= series(f,z,1001):
select(t -> coeff(S,z,t) <> 0, [$0..1000]); # Robert Israel, Oct 18 2015

f = EllipticTheta[2, 0, z^4]^3 + EllipticTheta[3, 0, z^4]^3; S = f + O[z]^200; Flatten[Position[CoefficientList[S, z], ?Positive] - 1] (* _Jean-François Alcover, Oct 23 2016, after Robert Israel *)

More terms from Sean A. Irvine, Oct 17 2015

A047703 Numbers that are the sum of 4 but no fewer positive cubes.

Original entry on oeis.org

4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 82, 88, 89, 93, 100, 107, 108, 119, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 219, 226, 233, 240, 245, 252, 254, 256, 259, 261, 266, 271, 278

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

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

Cf. A004215, A002376, A003327.

Select[Range[300], (pr = PowersRepresentations[#, 4, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

Original entry on oeis.org

1, 4, 2, -8, -4, 8, -8, -16, 6, 12, 8, -8, -8, 24, 0, -16, 12, 16, 10, -24, -8, 16, -24, -16, 8, 28, 8, -32, -16, 8, 0, -32, 6, 32, 16, -16, -12, 40, -24, -16, 24, 16, 16, -40, -8, 40, 0, -32, 24, 36, 10, -16, -24, 24, -32, -48, 0, 32, 24, -24, -16, 40, 0, -48, 12, 16, 16

Offset: 0

Michael Somos, Feb 28 2003

sign

Euler transform of period 4 sequence [4,-8,4,-3,...].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).
A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
      [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2015

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 25 2015, after Alois P. Heinz *)

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))

G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).

A134738 Cubes which are not the sum of three squares.

Original entry on oeis.org

343, 3375, 12167, 21952, 29791, 59319, 103823, 166375, 216000, 250047, 357911, 493039, 658503, 778688, 857375, 1092727, 1367631, 1404928, 1685159, 1906624, 2048383, 2460375, 2924207, 3442951, 3796416, 4019679, 4657463, 5359375

Offset: 1

Artur Jasinski, Nov 07 2007

nonn

This sequence was inspired by e-mail from Ray Chandler, Nov 07 2007

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

Cf. A004215, A134739.

N:= 10^10: # to get all terms <= N
sort([seq(seq(4^(3*i) * (8*j + 7)^3, j = 0 .. floor((N^(1/3)/4^i - 7)/8)), i = 0 .. floor(log[4](N^(1/3))))]); # Robert Israel, Dec 26 2017

b = Table[x^3, {x, 1, 300}]; a = {}; Do[Do[Do[AppendTo[a, (x^2 + y^2 + z^2)^3], {x, 0, 30}], {y, 0, 30}], {z, 0, 30}]; Union[a]; Complement[b, a] (*Artur Jasinski*)
Select[Range[200]^3,SquaresR[3,#]==0&] (* Harvey P. Dale, Feb 03 2015 *)

def A134738(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n)**3 # Chai Wah Wu, Feb 14 2025

a(n) = A004215(n)^3. - Ray Chandler, Jan 29 2009

Extended by Ray Chandler, Jan 29 2009

cp's OEIS Frontend

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

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Maple

Mathematica

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A125084 Cubes which have a partition as the sum of 3 squares.

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A278494 Primes p for which there does not exist any such integer k that k - A002828(k) = p.

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A319281 Numbers of the form 16^i*(16*j + 1).

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A343631 X-coordinate of the points following the 3D spiral defined in A343630.

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PARI

A343632 Y-coordinate of the points following the 3D spiral defined in A343630.

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PARI

A004014 Norms of vectors in the b.c.c. lattice.

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Maple

Mathematica

Extensions

A047703 Numbers that are the sum of 4 but no fewer positive cubes.

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Mathematica

A319281 Numbers of the form 16^i(16j + 1).