A001014 -id:A001014 - cp's OEIS frontend

A351617 Number of ways to write n as 11^w + x^2 + 2y^2 + 3z^2 + xyz, where w,x,y,z are nonnegative integers.

1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 4, 4, 3, 4, 1, 3, 5, 3, 5, 1, 5, 5, 1, 3, 4, 3, 6, 5, 5, 2, 4, 4, 3, 2, 8, 4, 5, 5, 5, 2, 4, 3, 5, 3, 5, 5, 5, 5, 7, 3, 5, 5, 4, 4, 3, 4, 8, 3, 8, 2, 6, 8, 3, 5, 4, 5, 10, 1, 5, 1, 4, 7, 4, 4, 7, 8, 11, 1, 3, 4, 5, 6, 7, 5, 6, 7, 7, 1, 5, 4, 10, 4, 7, 7, 4, 3, 7, 3, 8

Offset: 1

Zhi-Wei Sun, Mar 10 2022

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Let c be among 3, 4, 5, 7, 8. Then each positive integer n can be written as c^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
This has been verified for all n = 1..3*10^5.

			a(6) = 1 with 6 = 11^0 + 0^2 + 2*1^2 + 3*1^2 + 0*1*1.
a(24) = 1 with 24 = 11^1 + 1^2 + 2*0^2 + 3*2^2 + 1*0*2.
a(71) = 1 with 71 = 11^0 + 4^2 + 2*3^2 + 3*2^2 + 4*3*2.
a(89) = 1 with 89 = 11^0 + 4^2 + 2*6^2 + 3*0^2 + 4*6*0.
a(107) = 1 with 107 = 11^1 + 8^2 + 2*4^2 + 3*0^2 + 8*4*0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A001014, A351723, A351902, A352259, A352286.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-11^w-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,Log[11,n]},{z,0,Sqrt[(n-11^w)/3]},{y,0,Sqrt[(n-11^w-3z^2)/2]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A352259 Number of ways to write n as w^6 + x^2 + 2y^2 + 3z^2 + xyz, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 2, 3, 3, 3, 2, 3, 6, 4, 3, 2, 2, 5, 5, 5, 4, 3, 4, 2, 1, 5, 5, 4, 6, 5, 3, 3, 4, 5, 4, 5, 7, 5, 4, 5, 4, 3, 3, 3, 4, 3, 3, 5, 6, 7, 6, 5, 7, 6, 4, 4, 4, 7, 5, 4, 4, 3, 7, 5, 5, 6, 6, 10, 8, 3, 3, 4, 5, 8, 4, 9, 13, 12, 8, 2, 7, 10, 9, 10, 9, 7, 5, 3, 3, 8, 5, 10, 10, 6, 7, 8, 6, 10, 9, 11, 10

Offset: 0

Zhi-Wei Sun, Mar 10 2022

nonn

Conjecture 1: (i) a(n) > 0 for every n = 0,1,2,.... Moreover, 106, 744, 5469 and 331269 are the only nonnegative integers not in the set {w + x^2 + 2*y^2 + 3*z^2 + x*y*z: w = 0,1; x,y,z = 0,1,2,...}.
(ii) Let k be one of 4, 5, 6, 7. Then each n = 0,1,2,... can be written as 10*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
(iii) Let c be among 1, 3, 4, 6, 7, and let k be 4 or 5. Then every n = 0,1,2,... can be written as c*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
(iv) Each n = 0,1,2,... can be written as 9*w^4 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
Conjecture 2: Every n = 0,1,2,... can be written as 2*w^4 + 3*x^2 + y^2 + z^2 + x*y*z, where w,x,y,z are nonnegative integers.
We have verified Conjectures 1 and 2 for all n <= 10^5.

			a(24) = 1 with 24 = 0^6 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(106) = 1 with 106 = 2^6 + 1^2 + 2*2^2 + 3*3^2 + 1*2*3.

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

Cf. A000290, A000583, A000584, A001014, A351723, A351617, A351902, A352286.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-w^6-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,n^(1/6)},{z,0,Sqrt[(n-w^6)/3]},{y,0,Sqrt[(n-w^6-3z^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A377025 Squares and cubes that are not 6th powers.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025

Offset: 1

Chai Wah Wu, Oct 13 2024

Squares and cubes that cannot be written as both a square and a cube.
{A002760}\{A001014}.
A125643 minus the repeated terms.

Cf. A002760, A125643, A001014.

lim=2025;Select[Union[Range[Floor[lim^(1/2)]]^2,Range[Floor[lim^(1/3)]]^3],!IntegerQ[#^(1/6)]&] (* James C. McMahon, Oct 16 2024 *)

from math import isqrt
from sympy import integer_nthroot
def A377025(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: 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+(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n,n)

A069474 First differences of A069473.

Original entry on oeis.org

540, 2100, 5460, 11340, 20460, 33540, 51300, 74460, 103740, 139860, 183540, 235500, 296460, 367140, 448260, 540540, 644700, 761460, 891540, 1035660, 1194540, 1368900, 1559460, 1766940, 1992060, 2235540, 2498100, 2780460, 3083340

Offset: 0

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001014, A022522, A069473, A069475, A069476.

Equals 60 * A005898(n+1).

Differences[Table[(n + 1)^6 - n^6, {n, 0, 30}], 2] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = 120*n^3 + 540*n^2 + 900*n + 540.

G.f.: 60*(9 - x + 5*x^2 - x^3)/(1 - x)^4. [Bruno Berselli, Feb 25 2015]

Offset changed from 1 to 0 and added a(0)=540 by Bruno Berselli, Feb 25 2015

A069475 First differences of A069474, successive differences of (n+1)^6-n^6.

Original entry on oeis.org

1560, 3360, 5880, 9120, 13080, 17760, 23160, 29280, 36120, 43680, 51960, 60960, 70680, 81120, 92280, 104160, 116760, 130080, 144120, 158880, 174360, 190560, 207480, 225120, 243480, 262560, 282360, 302880, 324120, 346080, 368760, 392160, 416280

Offset: 0

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001014, A022522, A056107, A069473, A069474, A069476.

Table[360 n^2 + 1440 n + 1560, {n, 0, 40}] (* Bruno Berselli, Feb 25 2015 *)

a(n)=360*n^2+1440*n+1560 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 360*n^2 + 1440*n + 1560 = 120*A056107(n+2).

G.f.: 120*(13 - 11*x + 4*x^2)/(1 - x)^3. - Bruno Berselli, Feb 25 2015

Offset changed from 1 to 0 and added a(0)=1560 by Bruno Berselli, Feb 25 2015

A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

Original entry on oeis.org

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

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

			0.0013808358869717391630318541280158226106013963275654296802648025785307522...

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=6), page 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).

Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
  = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(2*n+1),
  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

Edited by M. F. Hasler, Apr 25 2021

A099764 a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

Original entry on oeis.org

0, 36, 576, 3600, 14400, 44100, 112896, 254016, 518400, 980100, 1742400, 2944656, 4769856, 7452900, 11289600, 16646400, 23970816, 33802596, 46785600, 63680400, 85377600, 112911876, 147476736, 190440000, 243360000, 308002500, 386358336

Offset: 0

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

			a(0) = 1^3 - 1^3 = 0;
a(1) = (1+3)^3 - (1^3+3^3) = 64 - 28 = 36;
a(2) = (1+3+5)^3 - (1^3+3^3+5^3) = 729 - 153 = 576;
a(3) = (1+3+5+7)^3 - (1^3+3^3+5^3+7^3) = 4096 - 496 = 3600;
a(4) = (1+3+5+7+9)^3 - (1^3+3^3+5^3+7^3+9^3) = 15625 - 1225 = 14400; etc. - _Philippe Deléham_, Mar 10 2014

Jolley, Summation of Series, Dover (1961).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001014, A002593.

List([1..30], n-> (n*(n^2-1))^2); # G. C. Greubel, Sep 03 2019

[n^2*(n+1)^2*(n+2)^2: n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

A099764:=n->n^2*(n+1)^2*(n+2)^2; seq(A099764(n), n=0..30); # Wesley Ivan Hurt, Mar 09 2014

Table[n^2*(n+1)^2*(n+2)^2, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
Times@@@Partition[Range[0,30]^2,3,1] (* Harvey P. Dale, Sep 02 2016 *)

vector(30, n, (n*(n^2-1))^2) \\ G. C. Greubel, Sep 03 2019

[(n*(n^2-1))^2 for n in (1..30)] # G. C. Greubel, Sep 03 2019

Sum_{n>=1} 1/a(n) = Pi^2/4-39/16 =  0.029901100272... [Jolley eq 241]
G.f.: 36*x*(1+x)*(1 +8*x +x^2)/(1-x)^7 . - R. J. Mathar, Oct 03 2011
a(n) = (Sum_{k=0..n} (2*k+1))^3 - Sum_{k=0..n} (2*k+1)^3. - Philippe Deléham, Mar 10 2014
a(n) = A001014(n+1) - A002593(n+1). - Philippe Deléham, Mar 10 2014
E.g.f.: exp(x)*x*(36+252*x+330*x^2+138*x^3+21*x^4+x^5). - Stefano Spezia, Sep 04 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 - 7/16. - Amiram Eldar, Jul 02 2020

A179124 Parameters n for which the elliptic curve y^2=x^3+n has rank 4.

Original entry on oeis.org

2089, 3391, 4481, 4910, 6856, 7057, 7954, 9052, 10333, 10636, 10942, 11321, 11665, 12092, 12742, 13191, 13897, 14129, 14668, 15193, 15501, 15641, 15661, 15689, 16306, 16376, 16649

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

See A031507 for the smallest n such that rank of the elliptic curve y^2=x^3+n is some given k.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151 (rank 0), A002153 (rank 1), A002155 (rank 2), A102833 (rank 3), A031507.

a(9)-a(27) from Seiichi Manyama, Jul 07 2019

A179127 Numbers n for which the order of Tate-Shafarevich group Ш (Sha) of the elliptic curve y^2=x^3+n is 4.

Original entry on oeis.org

123, 174, 214, 231, 286, 362, 383, 445, 487, 510, 527, 546, 566, 571, 608, 627, 669, 706, 718, 734, 741, 762, 805, 914, 942, 965, 970, 1019, 1042, 1059, 1075, 1131, 1155, 1166, 1189, 1203, 1210, 1230, 1236, 1245, 1287, 1320, 1355, 1392, 1397, 1410, 1411

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

For n<123 the order of the Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 1.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507.

Edited by N. J. A. Sloane, Jul 05 2010

A201217 Numbers such that (closest square) = (closest cube).

Original entry on oeis.org

0, 1, 2, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739

Offset: 1

Reinhard Zumkeller, Nov 28 2011

nonn

A061023(a(n)) = 0; A053187(a(n)) = A201053(a(n));

A001014 is a subsequence (6th powers).

Reinhard Zumkeller, Table of n, a(n) for n = 1..6051

import Data.List (elemIndices)
a201217 n = a201217_list !! (n-1)
a201217_list = elemIndices 0 a061023_list
-- Reinhard Zumkeller, Nov 28 2011

cp's OEIS Frontend

A351617 Number of ways to write n as 11^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

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A352259 Number of ways to write n as w^6 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

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A377025 Squares and cubes that are not 6th powers.

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A069474 First differences of A069473.

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A069475 First differences of A069474, successive differences of (n+1)^6-n^6.

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A085995 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+3.

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A099764 a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

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A179124 Parameters n for which the elliptic curve y^2=x^3+n has rank 4.

A351617 Number of ways to write n as 11^w + x^2 + 2y^2 + 3z^2 + xyz, where w,x,y,z are nonnegative integers.

A352259 Number of ways to write n as w^6 + x^2 + 2y^2 + 3z^2 + xyz, where w,x,y,z are nonnegative integers.