A056020 -id:A056020 - cp's OEIS frontend

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807

Offset: 1

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 9*X^3 + X^2 = Y^2.
The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010
The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - Bruno Berselli, May 15 2017

Jason Kimberley, Table of n, a(n) for n = 1..2108
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See A(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A205808 is the characteristic function.
Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - Jason Kimberley, Nov 09 2012
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

a:=func; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // Jason Kimberley, Nov 08 2012

readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2,n=0..39); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n,{n,-30,30}]] (* Harvey P. Dale, Dec 06 2013 *)

a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015

a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014
Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012

Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012

A129805 Primes congruent to +-1 mod 18.

Original entry on oeis.org

17, 19, 37, 53, 71, 73, 89, 107, 109, 127, 163, 179, 181, 197, 199, 233, 251, 269, 271, 307, 359, 379, 397, 431, 433, 449, 467, 487, 503, 521, 523, 541, 557, 577, 593, 613, 631, 647, 683, 701, 719, 739, 757, 773, 809, 811, 827, 829, 863, 881, 883, 919, 937, 953, 971, 991

Offset: 1

N. J. A. Sloane, May 22 2007

From Katherine E. Stange, Feb 03 2010: (Start)
Equivalently, primes p such that the smallest extension of F_p containing the cube roots of unity also contains the 9th roots of unity.
Equivalently, the primes p for which, if p^t = 1 mod 3, then p^t = 1 mod 9.
Equivalently, primes congruent to +/-1 modulo 9.
Membership or non-membership of the prime p in this sequence and sequence A002144 (primes congruent to 1 mod 4; equivalently, primes p such that the smallest extension of F_p containing the square roots of unity contains the 4th roots of unity) appear to determine the behavior of amicable pairs on the elliptic curve y^2 = x^3 + p (Silverman, Stange 2009). (End)
Primes in A056020. - Reinhard Zumkeller, Jan 07 2012
Primes congruent to (1,17) mod 18. - Vincenzo Librandi, Aug 14 2012
Equivalently, primes such that p^2 == 1 (mod 9). - M. F. Hasler, Apr 16 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emma Lehmer, On special primes, Pac. J. Math., 118 (1985), 471-478.
J. H. Silverman and K. E. Stange. Amicable pairs and aliquot cycles for elliptic curves, arxiv:0912.1831 [math.NT], 2009.

Cf. A000040, A010051.

Cf. A129806, A129807.

a129805 n = a129805_list !! (n-1)
a129805_list = [x | x <- a056020_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1300) | p mod 18 in {1, 17} ]; // Vincenzo Librandi, Aug 14 2012

Union[Join[Select[Range[-1, 3000, 18], PrimeQ], Select[Range[1, 3000, 18], PrimeQ]]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)
Select[Prime[Range[4000]],MemberQ[{1,17},Mod[#,18]]&] (* Vincenzo Librandi, Aug 14 2012 *)

select( {is_A129805(n)=n^2%9==1&&isprime(n)}, primes(199)) \\ M. F. Hasler, Apr 16 2022

A175886 Numbers that are congruent to {1, 12} mod 13.

Original entry on oeis.org

1, 12, 14, 25, 27, 38, 40, 51, 53, 64, 66, 77, 79, 90, 92, 103, 105, 116, 118, 129, 131, 142, 144, 155, 157, 168, 170, 181, 183, 194, 196, 207, 209, 220, 222, 233, 235, 246, 248, 259, 261, 272, 274, 285, 287, 298, 300, 311, 313, 324, 326, 337, 339, 350

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 13).

Bruno Berselli, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A091998, A113801, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A175887.

Cf. A195045 (partial sums).

a175886 n = a175886_list !! (n-1)
a175886_list = 1 : 12 : map (+ 13) a175886_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..350] | n mod 13 in [1, 12]]; // Bruno Berselli, Feb 29 2012

[(26*n+9*(-1)^n-13)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Select[Range[1, 350], MemberQ[{1, 12}, Mod[#, 13]]&] (* Bruno Berselli, Feb 29 2012 *)
CoefficientList[Series[(1 + 11 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{1,1,-1},{1,12,14},60] (* Harvey P. Dale, Oct 23 2015 *)

a(n)=(26*n+9*(-1)^n-13)/4 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(1+11*x+x^2)/((1+x)*(1-x)^2).
a(n) = (26*n+9*(-1)^n-13)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = a(n-2)+13.
a(n) = 13*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n>1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/13)*cot(Pi/13). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((26*x - 13)*exp(x) + 9*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/13).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/13)*cosec(Pi/13). (End)

A175887 Numbers that are congruent to {1, 14} mod 15.

Original entry on oeis.org

1, 14, 16, 29, 31, 44, 46, 59, 61, 74, 76, 89, 91, 104, 106, 119, 121, 134, 136, 149, 151, 164, 166, 179, 181, 194, 196, 209, 211, 224, 226, 239, 241, 254, 256, 269, 271, 284, 286, 299, 301, 314, 316, 329, 331, 344, 346, 359, 361, 374, 376, 389, 391, 404

Offset: 1

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 15).

Bruno Berselli, Table of n, a(n) for n = 1..10000.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.

Cf. A132240 (primes).

a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..450] | n mod 15 in [1,14]];

[(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Select[Range[1, 450], MemberQ[{1,14}, Mod[#, 15]]&]
CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)

a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).
a(n) = (30*n+11*(-1)^n-15)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.
a(n) = A047209(A047225(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - David Lovler, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)

A143548 Irregular triangle of numbers k < p^2 such that p^2 divides k^(p-1)-1, with p=prime(n).

Original entry on oeis.org

1, 1, 8, 1, 7, 18, 24, 1, 18, 19, 30, 31, 48, 1, 3, 9, 27, 40, 81, 94, 112, 118, 120, 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288, 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360

Offset: 1

T. D. Noe, Aug 24 2008

Row n begins with 1 and has prime(n)-1 terms. The first differences of each row are symmetric. For k > p^2, the solutions are just shifted by m*p^2 for m > 0. An open question is whether every integer appears in this sequence. For instance, 2 does not appear until the prime 1093 and 5 does not appear until the prime 20771.
For row n > 1, the sum of the terms in row n is (p-1)*p^2*(p+1)/2, which is A138416. - T. D. Noe, Aug 24 2008, corrected by Robert Israel, Sep 27 2016
Max Alekseyev points out that there is a much faster method of computing these numbers. Let p=prime(n) and let r be a primitive root of p (see A001918 and A060749). Then the terms in row n are r^(k*p) (mod p^2) for k=0..p-2. - T. D. Noe, Aug 26 2008
The numbers in this sequence are the bases to Euler pseudoprimes q, which are squares of prime numbers, such that n^((q-1)/2) == +-1 mod q. An exception is the first number 9 = 3*3, which is, following the strict definition in Crandall and Pomerance, no Fermat pseudoprime and hence no Euler pseudoprime. - Karsten Meyer, Jan 08 2011
For row n > 1, the sum is zero modulo p^2 (rows are antisymmetric due to Binomial Theorem). - Peter A. Lawrence, Sep 11 2016

			(2)   1,
(3)   1, 8,
(5)   1, 7, 18, 24,
(7)   1, 18, 19, 30, 31, 48,
(11)  1, 3, 9, 27, 40, 81, 94, 112, 118, 120,
(13)  1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168,
(17)  1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288,

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005

T. D. Noe, Rows n=1..50 of triangle, flattened

Cf. A039678, A056020, A056021, A056022, A056024, A056025, A056027, A056028, A056031, A056034, A056035, A096082, A138416.

f:= proc(n) local p,j,x;
  p:= ithprime(n);
  x:= numtheory:-primroot(p);
  op(sort([seq(x^(i*p) mod p^2, i=0..p-2)]))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 27 2016

Flatten[Table[p=Prime[n]; Select[Range[p^2], PowerMod[ #,p-1,p^2]==1&], {n,50}]] (* T. D. Noe, Aug 24 2008 *)
Flatten[Table[p=Prime[n]; r=PrimitiveRoot[p]; b=PowerMod[r,p,p^2]; Sort[NestList[Mod[b*#,p^2]&,1,p-2]], {n,50}]] (* Faster version from T. D. Noe, Aug 26 2008 *)

A332437 Decimal expansion of 2*cos(Pi/9).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 27 2020

This algebraic number called rho(9) of degree 3 = A055034(9) has minimal polynomial C(9, x) = x^3 - 3*x - 1 (see A187360).
rho(9) gives the length ratio diagonal/side of the smallest diagonal in the regular 9-gon.
The length ratio diagonal/side of the second smallest and the third smallest (or the largest) diagonal in the regular 9-gon are rho(9)^2 - 1 = A332438 - 1 and rho(9) + 1, respectively. - Mohammed Yaseen, Oct 31 2020

			rho(9) = 1.87938524157181676810821855464946293987241626852892926618...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Index entries for algebraic numbers, degree 3.

Cf. A019879, A055034, A056020, A156638, A187360, A214778, A215885, A332438.

RealDigits[2 * Cos[Pi/9], 10, 100][[1]] (* Amiram Eldar, Mar 27 2020 *)

2*cos(Pi/9) \\ Michel Marcus, Mar 28 2020

rho(9) = 2*cos(Pi/9).
Equals (-1)^(-1/9)*((-1)^(1/9) - i)*((-1)^(1/9) + i). - Peter Luschny, Mar 27 2020
Equals 2*A019879. - Michel Marcus, Mar 28 2020
Equals sqrt(A332438). - Mohammed Yaseen, Oct 31 2020
From Peter Bala, Oct 20 2021: (Start)
The zeros of x^3 - 3*x - 1 are r_1 = -2*cos(2*Pi/9), r_2 = -2*cos(4*Pi/9) and r_3 = -2*cos(8*Pi/9) = 2*cos(Pi/9).
The polynomial x^3 - 3*x - 1 is irreducible over Q (since it is irreducible mod 2) with discriminant equal to 3^4, a square. It follows that the Galois group of the number field Q(2*cos(Pi/9)) over Q is cyclic of order 3.
The mapping r -> 2 - r^2 cyclically permutes the zeros r_1, r_2 and r_3. The inverse cyclic permutation is given by r -> r^2 - r - 2.
The first differences r_1 - r_2, r_2 - r_3 and r_3 - r_1 are the zeros of the cyclic cubic polynomial x^3 - 9*x - 9 of discriminant 3^6.
First quotient relations:
r_1/r_2 = 1 + (r_3 - r_1); r_2/r_3 = 1 + (r_1 - r_2); r_3/r_1 = 1 + (r_2 - r_3);
r_2/r_1 = (r_3 - r_2) - 2; r_3/r_2 = (r_1 - r_3) - 2; r_1/r_3 = (r_2 - r_1) - 2;
r_1/r_2 + r_2/r_3 + r_3/r_1 = 3; r_2/r_1 + r_3/r_2 + r_1/r_3 = -6.
Thus the first quotients r_1/r_2, r_2/r_3 and r_3/r_1 are the zeros of the cyclic cubic polynomial x^3 - 3*x^2 - 6*x - 1 of discriminant 3^6. See A214778.
Second quotient relations:
(r_1*r_2)/(r_3^2) = 3*r_2 - 6*r_1 - 8, with two other similar relations by cyclically permuting the 3 zeros. The three second quotients are the zeros of the cyclic cubic polynomial x^3 + 24*x^2 + 3*x - 1 of discriminant 3^10.
(r_1^2)/(r_2*r_3) = 1 - 3*(r_2 + r_3), with two other similar relations by cyclically permuting the 3 zeros. (End)
Equals i^(2/9) + i^(-2/9). - Gary W. Adamson, Jun 25 2022
Equals Re((4+4*sqrt(3)*i)^(1/3)). - Gerry Martens, Mar 19 2024
From Amiram Eldar, Nov 22 2024: (Start)
Equals Product_{k>=1} (1 - (-1)^k/A056020(k)).
Equals 1 + Product_{k>=1} (1 + (-1)^k/A156638(k)). (End)

A195042 Concentric 9-gonal numbers.

Original entry on oeis.org

0, 1, 9, 19, 36, 55, 81, 109, 144, 181, 225, 271, 324, 379, 441, 505, 576, 649, 729, 811, 900, 991, 1089, 1189, 1296, 1405, 1521, 1639, 1764, 1891, 2025, 2161, 2304, 2449, 2601, 2755, 2916, 3079, 3249, 3421, 3600, 3781, 3969, 4159, 4356, 4555, 4761, 4969, 5184, 5401, 5625

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric enneagonal numbers or concentric nonagonal numbers.
A016766 and A069131 interleaved.
Partial sums of A056020. - Reinhard Zumkeller, Jan 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column 9 of A195040.

Cf. A016766, A069131, A077221, A195041, A195142, A195043.

a195042 n = a195042_list !! n
a195042_list = scanl (+) 0 a056020_list
-- Reinhard Zumkeller, Jan 07 2012

[(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* Harvey P. Dale, Nov 24 2019 *)

a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A060544(n+1). (End)
Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - Amiram Eldar, Jan 16 2023

A054966 Numbers that are congruent to {0, 1, 8} mod 9.

Original entry on oeis.org

0, 1, 8, 9, 10, 17, 18, 19, 26, 27, 28, 35, 36, 37, 44, 45, 46, 53, 54, 55, 62, 63, 64, 71, 72, 73, 80, 81, 82, 89, 90, 91, 98, 99, 100, 107, 108, 109, 116, 117, 118, 125, 126, 127, 134, 135, 136, 143, 144, 145, 152, 153, 154, 161, 162, 163, 170, 171, 172, 179, 180

Offset: 1

Henry Bottomley, May 24 2000

n == n^3 mod 9, so the iterated sum of the decimal digits of n and n^3 are equal.

H. I. Okagbue, M.O.Adamu, S.A. Bishop and A.A. Opanuga, Properties of Sequences Generated by Summing the Digits of Cubed Positive Integers, Indian Journal Of Natural Sciences, Vol. 6 / Issue 32 / October 2015

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

Cf. A047523. Complement of A275910.

[n : n in [0..200] | n mod 9 in [0, 1, 8]]; // Wesley Ivan Hurt, Jun 14 2016

A054966:=n->3*n-3+2*cos(2*n*Pi/3)+2*sin(2*n*Pi/3)/sqrt(3): seq(A054966(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[0, 200], MemberQ[{0, 1, 8}, Mod[#, 9]] &] (* Wesley Ivan Hurt, Jun 14 2016 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 8, 9}, 100] (* Vincenzo Librandi, Jun 15 2016 *)

G.f.: x^2*(1+7*x+x^2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 3*n-3+2*cos(2*n*Pi/3)+2*sin(2*n*Pi/3)/sqrt(3).
a(3k) = 9k-1, a(3k-1) = 9k-8, a(3k-2) = 9k-9. (End)
A008591 UNION A056020. - R. J. Mathar, Jul 19 2024
a(n) -a(n-1) = A105395(n+1), n>1. - R. J. Mathar, Jul 19 2024

A381319 Order of linear recurrence with constant coefficients of solutions of k satisfying k^(n-1) == 1 (mod n^2) for a given n.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 5, 2, 17, 2, 19, 2, 5, 2, 23, 2, 5, 2, 3, 4, 29, 2, 31, 2, 5, 2, 5, 2, 37, 2, 5, 2, 41, 2, 43, 2, 9, 2, 47, 2, 7, 2, 5, 4, 53, 2, 5, 2, 5, 2, 59, 2, 61, 2, 5, 2, 17, 6, 67, 2, 5, 4, 71, 2, 73, 2, 5, 4, 5, 2, 79, 2, 3, 2, 83, 2, 17, 2, 5, 2, 89

Offset: 2

Mike Sheppard, Feb 20 2025

nonn

For a given n, the solutions for k have the linear recurrence with constant coefficients k(m) = k(m-1) + k(m-(a(n)-1)) - k(m-a(n)), with order a(n). If a(n)=2 then the term k(m-1) appears twice and is k(m) = 2*k(m-1) - k(m-2).

Also, k(m) - k(m-(a(n)-1)) = n^2 = k(m-1) - k(m-a(n)), so all have nonhomogeneous linear recurrence of k(m) = k(m-(a(n)-1)) + n^2. Equivalently, k(m) = k(m-A063994(n)) + n^2, with order A063994(n). Thus, k(m) ~ (n^2 / A063994(n)) * m = (n^2 / (a(n)-1)) * m.

			For n=5 the congruence equation k^4 ==1 mod (5^2) has solutions of k (A056021) which satisfy k(m) = k(m-1) + k(m-4) - k(m-5), the order being 5, a(5)=5.
For n=9, k^8==1 mod (9^2) has solutions of k with recurrence k(m) = k(m-1) + k(m-2) - k(m-3), order 3, a(9)=3.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A063994, A056020 (n=3), A056021 (n=5), A056022 (n=7), A056024 (n=11), A056025 (n=13), A056028 (n=19), A056031 (n=23), A056034 (n=29), A056035 (n=31).

A381319[n_] := Times @@ GCD[FactorInteger[n][[All, 1]] - 1, n - 1] + 1;
Array[A381319, 100, 2] (* Paolo Xausa, Mar 05 2025 *)

a(n) = 1 + A063994(n).

a(p) = p if p is prime.

A056527 Numbers where iterated sum of digits of square settles down to a cyclic pattern (in fact 13, 16, 13, 16, ...).

Original entry on oeis.org

2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25, 29, 31, 32, 34, 38, 40, 41, 43, 47, 49, 50, 52, 56, 58, 59, 61, 65, 67, 68, 70, 74, 76, 77, 79, 83, 85, 86, 88, 92, 94, 95, 97, 101, 103, 104, 106, 110, 112, 113, 115, 119, 121, 122, 124, 128, 130, 131, 133, 137, 139, 140

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers == 2, 4, 5 or 7 mod 9, i.e. such that n^4 is not congruent to n^2 mod 9.

Numbers congruent to {2, 4, 5, 7} mod 9.

			a(1)=2 because iteration starts 2, 4, 7, 13, 16, 13, 16, ....

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

Cf. A004159 for sum of digits of square, A056020 where iteration settles to 1, A056020 where iteration settles to 9, also A056528, A056529. Unhappy numbers A031177 deal with iteration of square of sum of digits not settling to a single result.

Flatten[Table[9n+{2,4,5,7},{n,0,20}]] (* or *) LinearRecurrence[{1,0,0,1,-1},{2,4,5,7,11},100] (* Harvey P. Dale, Apr 05 2015 *)

Vec(x*(2 + 2*x + x^2 + 2*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^80)) \\ Colin Barker, Dec 19 2017

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Apr 05 2015
From Colin Barker, Dec 19 2017: (Start)
G.f.: x*(2 + 2*x + x^2 + 2*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = (-9 + (-1)^(1+n) - (3-3*i)*(-i)^n - (3+3*i)*i^n + 18*n) / 8 where i=sqrt(-1).
(End)

cp's OEIS Frontend

A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.

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A129805 Primes congruent to +-1 mod 18.

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A175886 Numbers that are congruent to {1, 12} mod 13.

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A175887 Numbers that are congruent to {1, 14} mod 15.

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A143548 Irregular triangle of numbers k < p^2 such that p^2 divides k^(p-1)-1, with p=prime(n).

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A332437 Decimal expansion of 2*cos(Pi/9).

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A132355 Numbers of the form 9h^2 + 2h, for h an integer.