A046530 -id:A046530 - cp's OEIS frontend

A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

1, 2, 3, 5, 5, 6, 13, 14, 18, 10, 11, 15, 25, 26, 15, 28, 17, 36, 37, 25, 39, 22, 23, 42, 35, 50, 81, 71, 29, 30, 61, 72, 33, 34, 65, 99, 73, 74, 75, 70, 41, 78, 85, 55, 90, 46, 47, 84, 112, 70, 51, 137, 53, 162, 55, 218, 111, 58, 59, 75, 121, 122, 288, 208, 125, 66, 133, 85, 69, 130, 71, 306, 145, 146, 105, 203, 143, 150, 157

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

A087786 includes pairs (x,y) with x>y (which are excluded from the present sequence).

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

Cf. A000086, A087786, A046530, A290731, A376202, A376203, A376755, A376756.

a(n) = sum(x=0, n-1, sum(y=x, n-1, Mod(x, n)^3 == Mod(y, n)^3)); \\ Michel Marcus, Oct 06 2024

from collections import Counter
def A376757(n): return sum(d*(d+1)>>1 for d in Counter(pow(x,3,n) for x in range(n)).values()) # Chai Wah Wu, Oct 06 2024

A133677 Integers k such that prime(k)(2prime(k)-1)/3 is an integer.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 13, 15, 16, 17, 20, 23, 24, 26, 28, 30, 32, 33, 35, 39, 40, 41, 43, 45, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 66, 69, 71, 72, 76, 77, 79, 81, 83, 86, 87, 89, 91, 92, 94, 96, 97, 98, 102, 103, 104, 107, 108, 109, 113, 116, 118, 119, 120, 123, 124, 126

Offset: 1

Roger L. Bagula, Dec 28 2007

nonn

Apart from the term "2", the same as A091177. - Stefan Steinerberger, Dec 29 2007
Numbers n such that the number of distinct residues r in the congruence x^3 == r (mod p) is equal to p where p = prime(n). See A046530. - Michel Lagneau, Sep 28 2016
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Feb 28 2021

			4 is not in the sequence since prime(4)*(2*prime(4) - 1)/3 = 7*(2*7 - 1)/3 = 7*13/3 = 91/3 is not an integer, but 5 is in the sequence since prime(5)*(2*prime(5) - 1)/3 = 11*(2*11 - 1)/3 = 11*21/3 = 11*7 = 77 is an integer. - _Michael B. Porter_, Sep 28 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046530, A091177, A133645.

for n from 1 to 126 do if((ithprime(n) mod 3) mod 2=0) then print(n) fi od; # Gary Detlefs, Dec 06 2011

Union[Table[If[IntegerQ[Prime[n]*(2*Prime[n] - 1)/3], n, {}], {n, 1, 100}]]
pnQ[n_]:=Module[{pn=Prime[n]},IntegerQ[(pn(2pn-1))/3]]; Select[Range[ 150], pnQ] (* Harvey P. Dale, Oct 02 2011 *)
Sort@ Join[{2}, Select[ Range@ 126, Mod[2*Prime[#], 3] == 1 &]] (* Robert G. Wilson v, Sep 28 2016 *)
Select[Range[126], IntegerQ[Prime[#]*(2 *Prime[#] - 1)/3] &] (* Robert Price, Apr 19 2025 *)

Integers k such that (prime(k) mod 3) mod 2 = 0. - Gary Detlefs, Dec 06 2011

A046633 Number of cubic residues mod 5^n.

Original entry on oeis.org

1, 5, 21, 101, 505, 2521, 12601, 63005, 315021, 1575101, 7875505, 39377521, 196887601, 984438005, 4922190021, 24610950101, 123054750505, 615273752521, 3076368762601, 15381843813005, 76909219065021, 384546095325101

Offset: 0

David W. Wilson

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Index entries for linear recurrences with constant coefficients, signature (5,0,1,-5).

A046633 := proc(n)
    5^(n+2)+2*op(1+modp(n,3),[3,15,13]) ;
    %/31 ;
end proc:
seq(A046633(n),n=0..20) ; # R. J. Mathar, Oct 08 2017

a[n_] := a[n] = Table[PowerMod[k, 3, 5^n], {k, 1, 5^n}] // Union // Length;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 10}]
(* or: *)
LinearRecurrence[{5, 0, 1, -5}, {1, 5, 21, 101}, 22] (* Jean-François Alcover, Nov 23 2017 *)

a(n)=(5^(n+2)+30)\31 \\ Charles R Greathouse IV, Jan 03 2013

a(n) = A046530(5^n). Conjecture: a(n)= +5*a(n-1) +a(n-3) -5*a(n-4) with g.f. ( 1-4*x^2-5*x^3 ) / ( (x-1)*(5*x-1)*(1+x+x^2) ). - R. J. Mathar, Feb 27 2011
The conjecture is correct. - Charles R Greathouse IV, Jan 03 2013
31*a(n) = 5^(n+2)+2*b(n), where b(n)=3 if n==0 (mod 3), b(n)=15 if n==1 (mod 3) and b(n)=13 if b(n)==2 (mod 3). - R. J. Mathar, Oct 08 2017

A046635 Number of cubic residues mod 7^n.

Original entry on oeis.org

1, 3, 15, 99, 689, 4817, 33713, 235987, 1651903, 11563315, 80943201, 566602401, 3966216801, 27763517603, 194344623215, 1360412362499, 9522886537489, 66660205762417, 466621440336913, 3266350082358387, 22864450576508703

Offset: 0

David W. Wilson

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n (arXiv:math.NT/0604465).
Index entries for linear recurrences with constant coefficients, signature (7,0,1,-7).

A046635 := proc(n)
    7^(n+2)+2*op(1+modp(n,3),[61,85,82]) ;
    %/171 ;
end proc:
seq(A046635(n),n=0..20) ; # R. J. Mathar, Oct 08 2017

LinearRecurrence[{7, 0, 1, -7}, {1, 3, 15, 99}, 21] (* Jean-François Alcover, Nov 24 2017 *)

a(n) = A046530(7^n).

a(n)= +7*a(n-1) +a(n-3) -7*a(n-4) with g.f. ( 1-4*x-6*x^2-7*x^3 ) / ( (x-1)*(7*x-1)*(1+x+x^2) ). - R. J. Mathar, Feb 27 2011

A046636 Number of cubic residues mod 8^n.

Original entry on oeis.org

1, 5, 37, 293, 2341, 18725, 149797, 1198373, 9586981, 76695845, 613566757, 4908534053, 39268272421, 314146179365, 2513169434917, 20105355479333, 160842843834661, 1286742750677285, 10293942005418277, 82351536043346213, 658812288346769701, 5270498306774157605, 42163986454193260837

Offset: 0

David W. Wilson

Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [broken link]
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [cached copy]
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A007583, A046530, A046630, A047853, A226308.

LinearRecurrence[{9, -8}, {1, 5}, 20] (* Jean-François Alcover, Jan 19 2019 *)

a(n) = (4*8^n + 3)/7.
a(n) = 8*a(n-1) - 3 (with a(0)=1). - Vincenzo Librandi, Nov 18 2010
From R. J. Mathar, Feb 28 2011: (Start)
a(n) = A046530(8^n) = A046630(3*n).
G.f.: (1-4*x)/((1-8*x)*(1-x)). (End)
a(n+1) = A226308(3*n+2). - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(4*exp(7*x) + 3)/7.
a(n) = 9*a(n-1) - 8*a(n-2).
a(n) = A047853(n+1)/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A046638 Number of cubic residues mod 10^n, or number of distinct n-digit endings of cubes.

Original entry on oeis.org

1, 10, 63, 505, 5050, 47899, 466237, 4662370, 46308087, 461504593, 4615045930, 46111077091, 460913873941, 4609138739410, 46086465166623, 460840040641225, 4608400406412250, 46083388790070379, 460830811531341997, 4608308115313419970, 46083004243912737927

Offset: 0

David W. Wilson

			a(1)=10 because a cube may end with any digit (10 possible combinations); a(2)=63 because a cube may end with 63 2-digit combinations (including leading zeros).
A cube may end with 63 different 2-digit combinations: 00, 01, 03, 04, 07, 08, 09, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99. Numbers ending with 14 say cannot be cubes. See also A075821, A075823. - _Zak Seidov_, Oct 18 2002

Alois P. Heinz, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,0,134,-1340,0,-1133,11330,0,1000,-10000).

Cf. A075821, A075823.

a(n)=(5^(n+2)+30)\31*((4<Charles R Greathouse IV, Jan 03 2013

a(n) = A046530(10^n) = A046630(n)*A046633(n). - R. J. Mathar, Feb 28 2011
a(n) ~ 100/217 * 10^n, so large terms start 460829493.... - Charles R Greathouse IV, Jan 03 2013
G.f.: -(10000*x^9+9000*x^8-5130*x^6-2357*x^5+259*x^3+37*x^2-1) / ((x-1)*(2*x-1)*(5*x-1)*(10*x-1)*(x^2+x+1)*(25*x^2+5*x+1)*(4*x^2+2*x+1)). - Alois P. Heinz, Jan 03 2013

Edited by N. J. A. Sloane, Oct 19 2008

A323704 Number of cubic residues (including 0) modulo the n-th prime.

Original entry on oeis.org

2, 3, 5, 3, 11, 5, 17, 7, 23, 29, 11, 13, 41, 15, 47, 53, 59, 21, 23, 71, 25, 27, 83, 89, 33, 101, 35, 107, 37, 113, 43, 131, 137, 47, 149, 51, 53, 55, 167, 173, 179, 61, 191, 65, 197, 67, 71, 75, 227, 77, 233, 239, 81, 251, 257, 263, 269, 91, 93, 281, 95, 293

Offset: 1

Florian Severin, Jan 24 2019

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046530, A000040, A236959.

Table[With[{p=Prime[n]},If[Mod[p,3]==1,1+(p-1)/3,p]],{n,80}] (* Harvey P. Dale, Feb 02 2025 *)

a(n) = my(p=prime(n)); sum(k=0, p-1, ispower(Mod(k,p), 3)); \\ Michel Marcus, Feb 26 2019

from sympy import prime
def a(n):
  p = prime(n)
  return len(set([x**3 % p for x in range(p)]))

If prime(n) - 1 = 3k then a(n) = k+1, otherwise a(n) = prime(n). (Cf. formula for A236959.)
a(n) = A236959(n) + 1.
a(n) = A046530(A000040(n)). - Rémy Sigrist, Jan 24 2019

A046632 Number of cubic residues mod 4^n.

Original entry on oeis.org

1, 3, 10, 37, 147, 586, 2341, 9363, 37450, 149797, 599187, 2396746, 9586981, 38347923, 153391690, 613566757, 2454267027, 9817068106, 39268272421, 157073089683, 628292358730, 2513169434917, 10052677739667, 40210710958666

Offset: 0

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4)

I:=[1, 3, 10, 37]; [n le 4 select I[n] else 4*Self(n-1)+Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

LinearRecurrence[{4,0,1,-4},{1,3,10,37},40] (* Vincenzo Librandi, Jun 22 2012 *)

G.f.: (-4x^3 - 2x^2 - x+1)/((1-4x)*(1-x^3)).
a(n) = A046530(4^n) = A046630(2n). - R. J. Mathar, Feb 27 2011
a(n) = 4*a(n-1) + a(n-3) - 4*a(n-4). - Vincenzo Librandi, Jun 22 2012

A282779 Period of cubes mod n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20, 21, 22, 23, 24, 25, 26, 9, 28, 29, 30, 31, 32, 33, 34, 35, 12, 37, 38, 39, 40, 41, 42, 43, 44, 15, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 56, 57, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 68, 69, 70, 71, 24, 73, 74, 75, 76, 77, 78, 79, 80, 27

Offset: 1

Ilya Gutkovskiy, Feb 21 2017

The length of the period of A000035 (n=2), A010872 (n=3), A109718 (n=4), A070471 (n=5), A010875 (n=6), A070472 (n=7), A109753 (n=8), A167176 (n=9), A008960 (n = 10), etc. (see also comment in A000578 from R. J. Mathar).
Conjecture: let a_p(n) be the length of the period of the sequence k^p mod n where p is a prime, then a_p(n) = n/p if n == 0 (mod p^2) else a_p(n) = n.
For example: sequence k^7 mod 98 gives 1, 30, 31, 18, 19, 48, 49, 50, 79, 80, 67, 68, 97, 0, 1, 30, 31, 18, 19, 48, 49, 50, 79, 80, 67, 68, 97, 0, ... (period 14), 7 is a prime, 98 == 0 (mod 7^2) and 98/7 = 14.

			a(9) = 3 because reading 1, 8, 27, 64, 125, 216, 343, 512, ... modulo 9 gives 1, 8, 0, 1, 8, 0, 1, 8, 0, ... with period length 3.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000035, A000578, A008960, A010872, A010875, A046530, A070471, A070472, A109718, A109753, A167176, A186646.

a[1] = 1; a[n_] := For[k = 1, True, k++, If[Mod[k^3, n] == 0 && Mod[(k + 1)^3 , n] == 1, Return[k]]]; Table[a[n], {n, 1, 81}]

Apparently: a(n) = 2*a(n-9) - a(n-18).

Empirical g.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 3*x^8 + 8*x^9 + 7*x^10 + 6*x^11 + 5*x^12 + 4*x^13 + 3*x^14 + 2*x^15 + x^16) / ((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2). - Colin Barker, Feb 21 2017

A046634 Number of cubic residues mod 6^n.

Original entry on oeis.org

1, 6, 9, 35, 210, 1083, 6253, 37518, 222705, 1331099, 7986594, 47871651, 287102581, 1722615486, 10334532969, 62003849075, 372023094450, 2232108315723, 13392560190013, 80355361140078, 482131358602785

Offset: 0

David W. Wilson

Index entries for linear recurrences with constant coefficients, signature (6,0,36,-216,0,-251,1506,0,216,-1296)

LinearRecurrence[{6,0,36,-216,0,-251,1506,0,216,-1296},{1,6,9,35,210,1083,6253,37518,222705,1331099},30] (* Harvey P. Dale, Mar 17 2023 *)

From R. J. Mathar, Feb 27 2011: (Start)
a(n) = A046530(6^n) = A046631(n)*A046630(n).
a(n) = +6*a(n-1) +36*a(n-3) -216*a(n-4) -251*a(n-6) +1506*a(n-7) +216*a(n-9) -1296*a(n-10).
G.f.: ( 1-27*x^2-55*x^3+795*x^5+690*x^6-2808*x^8-1296*x^9 ) / ( (x-1) *(6*x-1) *(3*x-1) *(2*x-1) *(1+x+x^2) *(4*x^2+2*x+1) *(9*x^2+3*x+1) ). (End)

cp's OEIS Frontend

A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

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A133677 Integers k such that prime(k)*(2*prime(k)-1)/3 is an integer.

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A046633 Number of cubic residues mod 5^n.

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A046635 Number of cubic residues mod 7^n.

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A046636 Number of cubic residues mod 8^n.

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A046638 Number of cubic residues mod 10^n, or number of distinct n-digit endings of cubes.

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PARI

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A323704 Number of cubic residues (including 0) modulo the n-th prime.

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A046632 Number of cubic residues mod 4^n.

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A133677 Integers k such that prime(k)(2prime(k)-1)/3 is an integer.