A002654 -id:A002654 - cp's OEIS frontend

A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

0, 1, 5, 25, 65, 325, 1105, 4225, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
In a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points", James R. Buddenhagen asked: Which circles have the property that they pass through more lattice points than any smaller circle? and he gave the terms 1, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed Fares. In the same thread Gerry Myerson mentioned the factorization into primes of the form 4*k+1.
Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1. - Amiram Eldar, Sep 12 2019
Indices of records of A004018. Apart from the first term, also indices of records of A002654. - Jianing Song, May 20 2021

Ray Chandler, Table of n, a(n) for n = 1..425 (first 97 terms from Ray Chandler, terms 98-365 from Amiram Eldar)
James Buddenhagen, Circle with 3 lattice points, thread in sci.math (May 2002)
Hugo Pfoertner, Construction of the sequences A071383, A071384, A071385

Cf. A000448, A048610, A052199, A071384, A071385, A230655, A300162. Subsequence of A054994 (excluding first term). Where records occur in A004018. See A088959 for circles with integer radius.

Indices of records of Sum_{d|n} kronecker(m, d): A230655 (m=-3), this sequence (m=-4), A279541 (m=-6).

my(v=list(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A054994

from math import prod
from sympy import isprime
primes_congruent_1_mod_4 = [5]
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
def generate_A054994():
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(r,exponents):
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0:
        yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
n,record,radius=1,1,0
print(radius, end="") # or record, for A071385
for radius,expo in generate_A054994():
    num_points = 4*prod((e+1) for e in expo)
    if num_points>record:
        record = num_points
        n += 1
        print (",", radius, end="") # or record, for A071385
        if n==27: break
print()
# Günter Rote, Sep 12 2023

For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can multiply the points x+iy for which x^2 + y^2 = N by either 2+i or 2-i to get two new sets of points X+iY for which X^2 + Y^2 = 5N. This strictly increases the number since it is easy to see that the two sets aren't the same. - J. H. Conway, Jun 04 2002

lim n ->infinity Log(a(n))/n = 1. [Conjectured by Benoit Cloitre, proved by J. H. Conway]

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)

A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

1, 8, 26, 64, 126, 208, 342, 512, 703, 1008, 1330, 1664, 2198, 2736, 3276, 4096, 4914, 5624, 6858, 8064, 8892, 10640, 12166, 13312, 15751, 17584, 18980, 21888, 24390, 26208, 29790, 32768, 34580, 39312, 43092, 44992, 50654, 54864, 57148

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Dirichlet convolution of A000578 = n^3 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A000578, A101455, A175572.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, this sequence, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

max = 40; s = Sum[n^3*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015, after Vladeta Jovovic *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(3*e+3) - s[p]^(e+1))/(p^3 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

a(n) = sumdiv(n, d, d^3*(((n/d) % 4)==1)) - sumdiv(n, d, d^3*(((n/d) % 4)==3)); \\ Michel Marcus, Feb 16 2015

G.f.: Sum_{n>=1} n^3*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(3*e+3) - A101455(p)^(e+1))/(p^3 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A175572. (End)
a(n) = Sum_{d|n} (n/d)^3*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

Offset changed from 0 to 1 by R. J. Mathar, Jul 15 2010

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 1, 3, 2, 1, 0, 2, 2, 0, 1, 0, 2, 0, 1, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 1, 1, 3, 2, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 3, 0, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 1, 2, 1, 0, 3, 2, 2, 0, 2, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^12 + 2*x^13 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 197, Entry 44.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002654, A008441, A019670, A122856, A122857, A122865, A125079.

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := SeriesCoefficient[ (-2 + EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 4, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -36, d)))}; /* Michael Somos, Jul 30 2006 */

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -36, p) * X))) [n])}; /* Michael Somos, Jul 30 2006 */

{a(n)=polcoeff(sum(m=0,n\6+1,(-1)^m*(x^(6*m+1)/(1-x^(6*m+1)+x*O(x^n)) + x^(6*m+5)/(1-x^(6*m+5)+x*O(x^n)))),n)} /* Paul D. Hanna */

Expansion of -1 + (theta_3(q)^2 + theta_3(q^3)^2) / 2 in powers of q. - Michael Somos, Jul 09 2013
From Michael Somos, Jul 30 2006: (Start)
Moebius transform is period 12 sequence [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, ...].
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1(mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3(mod 4). (End)
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -36, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -36, p) * p^-s)). - Michael Somos, Jun 24 2011
a(2*n) = a(3*n) = a(n). a(2*n + 1) = A125079(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).
2 * a(n) = A122857(n) unless n=0. - Michael Somos, Jul 09 2013
G.f.: Sum_{n>=0} (-1)^n*( x^(6*n+1)/(1-x^(6*n+1)) + x^(6*n+5)/(1-x^(6*n+5)) ). - Paul D. Hanna, Dec 14 2011
G.f.: x/(1-x) + x^5/(1-x^5) - x^7/(1-x^7) - x^11/(1-x^11) + x^13/(1-x^13) + x^17/(1-x^17) --++ ...
a(n) = A002654(n) + A002654(3*n). - Michael Somos, Jan 25 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Nov 17 2023

A321829 a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

Original entry on oeis.org

1, 32, 242, 1024, 3126, 7744, 16806, 32768, 58807, 100032, 161050, 247808, 371294, 537792, 756492, 1048576, 1419858, 1881824, 2476098, 3201024, 4067052, 5153600, 6436342, 7929856, 9768751, 11881408, 14290100, 17209344, 20511150, 24207744, 28629150, 33554432, 38974100, 45435456, 52535556

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455, A175570.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, this sequence, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, # ^5 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(5*e+5) - s[p]^(e+1))/(p^5 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321829(n)=factorback(apply(f->f[1]^(5*f[2]+5)\/(f[1]^5+f[1]%4-2),Col(factor(n)))), [1..40]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(5e+5)/(p^5 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = A175570. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^5*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321830 a(n) = Sum_{d|n, n/d==1 mod 4} d^6 - Sum_{d|n, n/d==3 mod 4} d^6.

Original entry on oeis.org

1, 64, 728, 4096, 15626, 46592, 117648, 262144, 530713, 1000064, 1771560, 2981888, 4826810, 7529472, 11375728, 16777216, 24137570, 33965632, 47045880, 64004096, 85647744, 113379840, 148035888, 190840832, 244156251, 308915840, 386889776

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455, A258814.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, this sequence, A321831, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^6 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(6*e+6) - s[p]^(e+1))/(p^6 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321830(n)=factorback(apply(f->f[1]^(6*f[2]+6)\/(f[1]^6+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(6e+6)/(p^6 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = 61*Pi^7/184320 (A258814). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^6*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321831 a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

Original entry on oeis.org

1, 128, 2186, 16384, 78126, 279808, 823542, 2097152, 4780783, 10000128, 19487170, 35815424, 62748518, 105413376, 170783436, 268435456, 410338674, 611940224, 893871738, 1280016384, 1800262812, 2494357760, 3404825446, 4584374272, 6103593751

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455, A258815.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^7 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = A258815. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^7*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

Original entry on oeis.org

1, 256, 6560, 65536, 390626, 1679360, 5764800, 16777216, 43040161, 100000256, 214358880, 429916160, 815730722, 1475788800, 2562506560, 4294967296, 6975757442, 11018281216, 16983563040, 25600065536, 37817088000, 54875873280, 78310985280, 110058536960, 152588281251, 208827064832

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455, A258816.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, this sequence, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^8 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(8*e+8) - s[p]^(e+1))/(p^8 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321832(n)=factorback(apply(f->f[1]^(8*f[2]+8)\/(f[1]^8+f[1]%4-2),Col(factor(n)))), [1..50]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(8e+8)/(p^8 + (p mod 4) - 2)). (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = 277*Pi^9/8257536 (A258816). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^8*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.

Original entry on oeis.org

1, 512, 19682, 262144, 1953126, 10077184, 40353606, 134217728, 387400807, 1000000512, 2357947690, 5159518208, 10604499374, 20661046272, 38441425932, 68719476736, 118587876498, 198349213184, 322687697778, 512000262144, 794239673292

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, this sequence, A321834, A321835, A321836.

f:= n ->
mul(piecewise(t[1]=2,2^(9*t[2]), t[1] mod 4 = 1, (t[1]^(9*(t[2]+1))-1)/(t[1]^9-1), (t[1]^(9*(t[2]+1))+(-1)^t[2])/(t[1]^9+1)), t = ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Nov 26 2018

s[n_,r_] := DivisorSum[n, #^9 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(9*e+9) - s[p]^(e+1))/(p^9 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^9)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Robert Israel, Nov 26 2018: (Start) a(2^m) = 2^(9*m).
For prime p == 1 (mod 4), a(p^m) = (p^(9(m+1))-1)/(p^9-1).
For prime p == 3 (mod 4), a(p^m) = (p^(9(m+1))+(-1)^m)/(p^9+1). (End)
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - A101455(p)^(e+1))/(p^9 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = beta(10) = 0.99998316402... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^9*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

Original entry on oeis.org

1, 1024, 59048, 1048576, 9765626, 60465152, 282475248, 1073741824, 3486725353, 10000001024, 25937424600, 61916315648, 137858491850, 289254653952, 576640684048, 1099511627776, 2015993900450, 3570406761472, 6131066257800, 10240001048576

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

cp's OEIS Frontend

A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A321829 a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321830 a(n) = Sum_{d|n, n/d==1 mod 4} d^6 - Sum_{d|n, n/d==3 mod 4} d^6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321831 a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula