A381319 -id:A381319 - cp's OEIS frontend

A056020 Numbers that are congruent to +-1 mod 9.

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

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

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A056021 Numbers k such that k^4 == 1 (mod 5^2).

Original entry on oeis.org

1, 7, 18, 24, 26, 32, 43, 49, 51, 57, 68, 74, 76, 82, 93, 99, 101, 107, 118, 124, 126, 132, 143, 149, 151, 157, 168, 174, 176, 182, 193, 199, 201, 207, 218, 224, 226, 232, 243, 249, 251, 257, 268, 274, 276, 282, 293, 299, 301, 307, 318, 324, 326, 332, 343, 349

Offset: 1

Robert G. Wilson v, Jun 08 2000

Numbers congruent to {1, 7, 18, 24} mod 25.

These terms (apart from 1) are tetration bases characterized by a constant convergence speed strictly greater than 1 (see A317905). - Marco Ripà, Jan 25 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Colin Barker)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A056022, A056024, A056025, A056026, A056027, A056028, A056031, A056034, A056035, A317905.

Cf. A381319 (general case mod n^2)

Select[ Range[ 400 ], PowerMod[ #, 4, 25 ]==1& ]
(* or *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 7, 18, 24, 26}, 100] (* Mike Sheppard, Feb 18 2025 *)

a(n) = (-25 - (-1)^n + (9-9*I)*(-I)^n + (9+9*I)*I^n + 50*n) / 8 \\ Colin Barker, Oct 16 2015

Vec(x*(x^2+3*x+1)^2/((1+x)*(x^2+1)*(x-1)^2) + O(x^100)) \\ Colin Barker, Oct 16 2015

for(n=0, 1e3, if(n^4 % 5^2 == 1, print1(n", "))) \\ Altug Alkan, Oct 16 2015

isok(k) = Mod(k, 25)^4 == 1; \\ Michel Marcus, Jun 30 2021

G.f.: x*(x^2+3*x+1)^2 / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 25 2011
a(n) = (-25 - (-1)^n + (9-9*i)*(-i)^n + (9+9*i)*i^n + 50*n) / 8, where i = sqrt(-1). - Colin Barker, Oct 16 2015
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
a(n) ~ (5^2/4)*n. (End)

A056022 Numbers k such that k^6 == 1 (mod 7^2).

Original entry on oeis.org

1, 18, 19, 30, 31, 48, 50, 67, 68, 79, 80, 97, 99, 116, 117, 128, 129, 146, 148, 165, 166, 177, 178, 195, 197, 214, 215, 226, 227, 244, 246, 263, 264, 275, 276, 293, 295, 312, 313, 324, 325, 342, 344, 361, 362, 373, 374, 391, 393, 410, 411, 422, 423, 440, 442

Offset: 1

Robert G. Wilson v, Jun 08 2000

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

Cf. A056021, A056024, A056025, A056026, A056027, A056028, A056031, A056034, A056035.

Cf. A381319 (general case mod n^2).

Select[ Range[ 500 ], PowerMod[ #, 6, 49 ]==1& ]
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 18, 19, 30, 31, 48, 50}, 61] (* Mike Sheppard, Feb 18 2025 *)

isok(k) = Mod(k, 49)^6 == 1; \\ Michel Marcus, Jun 30 2021

From Mike Sheppard, Feb 18 2025 : (Start)
a(n) = a(n-1) + a(n-6) - a(n-7).
a(n) = a(n-6) + 7^2.
a(n) ~ (7^2/6)*n.
G.f.: (1 + x*(17 + x + 11*x^2 + x^3 + 17*x^4 + x^5))/(1 - x - x^6 + x^7). (End)

A056024 Numbers k such that k^10 == 1 (mod 11^2).

Original entry on oeis.org

1, 3, 9, 27, 40, 81, 94, 112, 118, 120, 122, 124, 130, 148, 161, 202, 215, 233, 239, 241, 243, 245, 251, 269, 282, 323, 336, 354, 360, 362, 364, 366, 372, 390, 403, 444, 457, 475, 481, 483, 485, 487, 493, 511, 524, 565, 578, 596, 602, 604, 606, 608, 614, 632

Offset: 1

Robert G. Wilson v, Jun 08 2000

nonn

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

Cf. A056021, A056022, A056025, A056026, A056027, A056028, A056031, A056034, A056035.

Cf. A381319 (general case mod n^2).

Select[ Range[ 800 ], PowerMod[ #, 10, 121 ]==1& ]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 3, 9, 27, 40, 81, 94, 112, 118, 120, 122}, 65] (* Mike Sheppard, Feb 19 2025 *)

From Mike Sheppard, Feb 19 2025: (Start)
a(n) = a(n-1) + a(n-10) - a(n-11).
a(n) = a(n-10) + 11^2.
a(n) ~ (11^2/10)*n. (End)

A056025 Numbers k such that k^12 == 1 (mod 13^2).

Original entry on oeis.org

1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 170, 188, 191, 192, 239, 249, 258, 268, 315, 316, 319, 337, 339, 357, 360, 361, 408, 418, 427, 437, 484, 485, 488, 506, 508, 526, 529, 530, 577, 587, 596, 606, 653, 654, 657, 675, 677, 695, 698, 699, 746

Offset: 1

Robert G. Wilson v, Jun 08 2000

From 19 to 168 inclusive, these are the numbers that 'fool' the strong pseudoprimality test described in Wilf (1986) in regard to determining whether 169 is composite. - Alonso del Arte, Feb 05 2012

Herbert S. Wilf, Algorithms and Complexity, Englewood Cliffs, New Jersey: Prentice-Hall, 1986, pp. 158-160.

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

Cf. A056021, A056022, A056024, A056026, A056027, A056028, A056031, A056034, A056035.

Cf. A381319 (general case mod n^2).

Select[ Range[ 800 ], PowerMod[ #, 12, 169 ]==1& ]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168, 170}, 56] (* Mike Sheppard, Feb 19 2025 *)

is(k)=Mod(k,169)^12==1 \\ Charles R Greathouse IV, Feb 07 2018

From Mike Sheppard, Feb 19 2025 : (Start)
a(n) = a(n-1) + a(n-12) - a(n-13).
a(n) = a(n-12) + 13^2.
a(n) ~ (13^2/12)*n. (End)

Definition corrected by T. D. Noe, Aug 23 2008

A056028 Numbers k such that k^18 == 1 (mod 19^2).

Original entry on oeis.org

1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360, 362, 389, 415, 423, 429, 430, 460, 477, 488, 595, 606, 623, 653, 654, 660, 668, 694, 721, 723, 750, 776, 784, 790, 791, 821, 838, 849, 956, 967, 984, 1014, 1015, 1021, 1029

Offset: 1

Robert G. Wilson v, Jun 08 2000

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

Cf. A056021, A056022, A056024, A056025, A056026, A056027.

Cf. A381319 (general case mod n^2).

x=19; Select[ Range[ 1250 ], PowerMod[ #, x-1, x^2 ]==1& ]

isok(n) = Mod(n, 19^2)^18 == 1; \\ Michel Marcus, Feb 12 2017

Vec(x*(x^18 +27*x^17 +26*x^16 +8*x^15 +6*x^14 +x^13 +30*x^12 +17*x^11 +11*x^10 +107*x^9 +11*x^8 +17*x^7 +30*x^6 +x^5 +6*x^4 +8*x^3 +26*x^2 +27*x +1) / (x^19 -x^18 -x +1) + O(x^100)) \\ Colin Barker, Feb 12 2017

a(n) = a(n-1) + a(n-18) - a(n-19). - Matthew House, Feb 12 2017
G.f.: x*(x^18 +27*x^17 +26*x^16 +8*x^15 +6*x^14 +x^13 +30*x^12 +17*x^11 +11*x^10 +107*x^9 +11*x^8 +17*x^7 +30*x^6 +x^5 +6*x^4 +8*x^3 +26*x^2 +27*x +1) / (x^19 -x^18 -x +1). - Colin Barker, Feb 12 2017
From Mike Sheppard, Feb 20 2025: (Start)
a(n) = a(n-18) + 19^2.
a(n) ~ (19^2/18)*n. (End)

A056031 Numbers k such that k^22 == 1 (mod 23^2).

Original entry on oeis.org

1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528, 530, 557, 571, 592, 647, 659, 699, 706, 724, 784, 792, 795, 803, 863, 881, 888, 928, 940, 995, 1016, 1030, 1057, 1059, 1086, 1100, 1121, 1176, 1188

Offset: 1

Robert G. Wilson v, Jun 08 2000

nonn

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

Cf. A056021, A056022, A056024, A056025, A056026, A056027, A056028, A056034, A056035.

Cf. A381319 (general case mod n^2).

x=23; Select[ Range[ 2000 ], PowerMod[ #, x-1, x^2 ]==1& ]

From Mike Sheppard, Feb 20 2025 : (Start)
a(n) = a(n-1) + a(n-22) - a(n-23).
a(n) = a(n-22) + 23^2.
a(n) ~ (23^2/22)*n. (End)

A056034 Numbers k such that k^28 == 1 (mod 29^2).

Original entry on oeis.org

1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840, 842, 855, 882, 901, 904, 978, 1031, 1037, 1062, 1077, 1108, 1111, 1215, 1257, 1266, 1308, 1412, 1415, 1446, 1461, 1486, 1492

Offset: 1

Robert G. Wilson v, Jun 08 2000

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

Cf. A056021, A056022, A056024, A056025, A056026, A056027, A056028, A056031, A056035.

Cf. A381319 (general case mod n^2).

x=29; Select[ Range[ 2000 ], PowerMod[ #, x-1, x^2 ]==1& ]

isok(k) = Mod(k, 29^2)^28 == 1; \\ Michel Marcus, Apr 10 2025

From Mike Sheppard, Feb 20 2025 : (Start)
a(n) = a(n-1) + a(n-28) - a(n-29).
a(n) = a(n-28) + 29^2.
a(n) ~ (29^2/28)*n. (End)

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A056020 Numbers that are congruent to +-1 mod 9.

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A056021 Numbers k such that k^4 == 1 (mod 5^2).

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A056022 Numbers k such that k^6 == 1 (mod 7^2).

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A056024 Numbers k such that k^10 == 1 (mod 11^2).

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A056025 Numbers k such that k^12 == 1 (mod 13^2).

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A056028 Numbers k such that k^18 == 1 (mod 19^2).

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A056031 Numbers k such that k^22 == 1 (mod 23^2).

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