A001017 -id:A001017 - cp's OEIS frontend

A306240 Number of ways to write n as x^9 + y^3 + z(z+1) + w(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w.

1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 5, 4, 3, 2, 1, 1, 2, 4, 4, 3, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 6, 4, 2, 2, 2, 2, 3, 5, 5, 4, 4, 4, 4, 2, 1, 3, 4, 5, 5, 3, 2, 2, 2, 3, 4, 4, 5, 4, 2, 4, 6, 5, 2, 2, 3, 4, 6, 6, 4, 4, 5, 3, 3, 6, 6, 4, 3, 3, 3, 3, 3, 5, 7, 6, 5, 3, 3, 4, 3, 5, 6, 4, 3, 4, 4, 3, 5, 6

Offset: 0

Zhi-Wei Sun, Jan 31 2019

nonn

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 11, 17, 18, 47, 108, 109, 234, 359. Also, any nonnegative integer can be written as x^6 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2.

We have verified a(n) > 0 for all n = 0..2*10^7.

			a(11) = 1 with 11 = 1^9 + 2^3 + 0*1 + 1*2.
a(18) = 1 with 18 = 0^9 + 0^3 + 2*3 + 3*4.
a(109) = 1 with 109 = 1^9 + 4^3 + 1*2 + 6*7.
a(234) = 1 with 234 = 0^9 + 6^3 + 2*3 + 3*4.
a(359) = 1 with 359 = 1^9 + 2^3 + 10*11 + 15*16.
a(1978) = 3 with 1978 = 2^9 + 2^3 + 26*27 + 27*28 = 2^9 + 6^3 + 19*20 + 29*30 = 2^9 + 6^3 + 24*25 + 25*26.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000578, A001014, A001017, A002378, A262813, A262815, A270488, A306225, A306227, A306239.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
tab={};Do[r=0;Do[If[TQ[n-x^9-y^3-z(z+1)],r=r+1],{x,0,Min[2,n^(1/9)]},{y,0,(n-x^9)^(1/3)},{z,0,(Sqrt[2(n-x^9-y^3)+1]-1)/2}];tab=Append[tab,r],{n,0,100}];Print[tab]

A344336 Number of divisors of n^9.

Original entry on oeis.org

1, 10, 10, 19, 10, 100, 10, 28, 19, 100, 10, 190, 10, 100, 100, 37, 10, 190, 10, 190, 100, 100, 10, 280, 19, 100, 28, 190, 10, 1000, 10, 46, 100, 100, 100, 361, 10, 100, 100, 280, 10, 1000, 10, 190, 190, 100, 10, 370, 19, 190, 100, 190, 10, 280, 100, 280, 100, 100, 10, 1900, 10, 100

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=9 of A343656.

Cf. A000005, A001017, A344337 (9^omega(n)).

Table[DivisorSigma[0, n^9], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^9);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 9*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 9^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 9^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 8*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001017(n)).
Multiplicative with a(p^e) = 9*e+1.
a(n) = Sum_{d|n} 9^omega(d).
G.f.: Sum_{k>=1} 9^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 8/p^s). - Vaclav Kotesovec, Aug 19 2021

A004886 Numbers that are the sum of at most 2 positive 9th powers.

Original entry on oeis.org

0, 1, 2, 512, 513, 1024, 19683, 19684, 20195, 39366, 262144, 262145, 262656, 281827, 524288, 1953125, 1953126, 1953637, 1972808, 2215269, 3906250, 10077696, 10077697, 10078208, 10097379, 10339840, 12030821, 20155392, 40353607, 40353608

Offset: 1

N. J. A. Sloane

nonn

Zhuorui He, Table of n, a(n) for n = 1..10000

Cf. A001017 (9th powers), A003391 (sum of 2).

lista(nn) = setbinop((x,y)->x^9+y^9, [0..nn]); \\ Michel Marcus, Jul 02 2025

def A004886_upto(n):
  a=set()
  for i in range(n):
    if 2*(i**9)>n: break
    for j in range(i,n):
      if i**9+j**9<=n: a.add(i**9+j**9)
      else: break
  return sorted(a) # Zhuorui He, Jun 30 2025

A016773 a(n) = (3*n)^9.

Original entry on oeis.org

0, 19683, 10077696, 387420489, 5159780352, 38443359375, 198359290368, 794280046581, 2641807540224, 7625597484987, 19683000000000, 46411484401953, 101559956668416, 208728361158759, 406671383849472

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A001017 (n^9).

[(3*n)^9: n in [0..20]]; // Vincenzo Librandi, May 09 2011

A016917 a(n) = (6*n)^9.

Original entry on oeis.org

0, 10077696, 5159780352, 198359290368, 2641807540224, 19683000000000, 101559956668416, 406671383849472, 1352605460594688, 3904305912313344, 10077696000000000, 23762680013799936, 51998697814228992

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A001017, A008588.

[(6*n)^9: n in [0..30]]; // Vincenzo Librandi, May 03 2011

A016989 a(n) = (7*n)^9.

Original entry on oeis.org

0, 40353607, 20661046784, 794280046581, 10578455953408, 78815638671875, 406671383849472, 1628413597910449, 5416169448144896, 15633814156853823, 40353607000000000, 95151694449171437, 208215748530929664

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A001017.

[(7*n)^9: n in [0..20]]; // Vincenzo Librandi, Jun 18 2011

(7 Range[0, 20])^9 (* Wesley Ivan Hurt, May 26 2024 *)

From Wesley Ivan Hurt, May 26 2024: (Start)
a(n) = 40353607 * A001017(n).
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10).
G.f.: 40353607*x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(x - 1)^10. (End)

A017313 a(n) = (10*n + 3)^9.

Original entry on oeis.org

19683, 10604499373, 1801152661463, 46411484401953, 502592611936843, 3299763591802133, 15633814156853823, 58871586708267913, 186940255267540403, 520411082988487293, 1304773183829244583, 3004041937984268273

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A001017 (n^9), A017305 (10n+3).

[(10*n+3)^9: n in [0..15]]; // Vincenzo Librandi, Jul 31 2011

(10*Range[0,20]+3)^9 (* or *) LinearRecurrence[ {10,-45,120,-210,252,-210,120,-45,10,-1},{19683,10604499373,1801152661463,46411484401953,502592611936843,3299763591802133,15633814156853823,58871586708267913,186940255267540403,520411082988487293},30] (* Harvey P. Dale, Sep 14 2013 *)

for n in range(0, 15): print((10*n + 3)**9, end=", ") # Stefano Spezia, Oct 20 2018

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10); a(0)=19683, a(1)=10604499373, a(2)=1801152661463, a(3)=46411484401953, a(4)=502592611936843, a(5)=3299763591802133, a(6)=15633814156853823, a(7)=58871586708267913, a(8)=186940255267540403, a(9)=520411082988487293. - Harvey P. Dale, Sep 14 2013

A017361 a(n) = (10*n + 7)^9.

Original entry on oeis.org

40353607, 118587876497, 7625597484987, 129961739795077, 1119130473102767, 6351461955384057, 27206534396294947, 95151694449171437, 285544154243029527, 760231058654565217, 1838459212420154507, 4108400332687853397

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A017353 (10n+7), A001017 (n^9).

[(10*n+7)^9: n in [0..20]]; // Vincenzo Librandi, Aug 30 2011

(10*Range[0,30]+7)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{40353607,118587876497,7625597484987,129961739795077,1119130473102767,6351461955384057,27206534396294947,95151694449171437,285544154243029527,760231058654565217},30] (* Harvey P. Dale, Dec 28 2011 *)

vector(20, n, n--; (10*n+7)^9) \\ G. C. Greubel, Nov 10 2018

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10); a(0)=40353607, a(1)=118587876497, a(2)=7625597484987, a(3)=129961739795077, a(4)=1119130473102767, a(5)=6351461955384057, a(6)=27206534396294947, a(7)=95151694449171437, a(8)=285544154243029527, a(9)=760231058654565217. - Harvey P. Dale, Dec 28 2011

A050756 Ninth powers containing no pair of consecutive equal digits.

Original entry on oeis.org

0, 1, 512, 19683, 1953125, 40353607, 387420489, 2357947691, 5159780352, 68719476736, 198359290368, 3814697265625, 5429503678976, 14507145975869, 165216101262848, 327381934393961, 618121839509504, 18014398509481984

Offset: 0

Patrick De Geest, Sep 15 1999

Cf. A043096, A001017, A050748.

Select[Range[0,90]^9,FreeQ[Differences[IntegerDigits[#]],0]&] (* Harvey P. Dale, Dec 27 2011 *)

A017577 a(n) = (12n+4)^9.

Original entry on oeis.org

262144, 68719476736, 10578455953408, 262144000000000, 2779905883635712, 18014398509481984, 84590643846578176, 316478381828866048, 1000000000000000000, 2773078757450186752, 6930988311686938624, 15916595351771938816, 34068690316840665088, 68719476736000000000

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001017, A013667, A016785, A017569.

(12*Range[0,30]+4)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{262144,68719476736,10578455953408,262144000000000,2779905883635712,18014398509481984,84590643846578176,316478381828866048,1000000000000000000,2773078757450186752},40] (* Harvey P. Dale, Sep 07 2018 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A001017(A017569(n)) = A017569(n)^9.
a(n) = 262144 * A016785(n).
Sum_{n>=0} 1/a(n) = 809*Pi^9/(7313988648960*sqrt(3)) + 9841*zeta(9)/5159780352. (End)

More terms from Amiram Eldar, Jul 14 2024

cp's OEIS Frontend

A306240 Number of ways to write n as x^9 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A344336 Number of divisors of n^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A004886 Numbers that are the sum of at most 2 positive 9th powers.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

A016773 a(n) = (3*n)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

A016917 a(n) = (6*n)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

A016989 a(n) = (7*n)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A017313 a(n) = (10*n + 3)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A017361 a(n) = (10*n + 7)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A306240 Number of ways to write n as x^9 + y^3 + z(z+1) + w(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w.