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A128394 Numbers k such that k^2 divides 14^k - 1.

1, 13, 2041, 24355253, 249302027, 16772956369, 39665616523, 388885239223, 2974921088191, 3487599163841, 61054982558011, 200151688351277, 473329801968959

Offset: 1

Alexander Adamchuk, Mar 01 2007

13 divides all except the first term.

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a(4)-a(5) from Farideh Firoozbakht, Mar 05 2007

a(6)-a(13) from Max Alekseyev, May 06 2010

A128396 Numbers k such that k^2 divides 16^k-1.

Original entry on oeis.org

1, 3, 5, 15, 21, 39, 55, 105, 155, 165, 195, 205, 273, 465, 609, 615, 903, 915, 1155, 1265, 1365, 1705, 2067, 2145, 2255, 2265, 2373, 2667, 3045, 3081, 3255, 3795, 4305, 4515, 4895, 4965, 5115, 6045, 6123, 6355, 6405, 6765, 7077, 7455, 7917, 7995, 10065

Offset: 1

Alexander Adamchuk, Mar 01 2007

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

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Select[Range[11000],Divisible[16^#-1,#^2]&] (* Harvey P. Dale, Dec 14 2010 *)
Join[{1},Select[Range[11000],PowerMod[16,#,#^2]==1&]] (* Harvey P. Dale, Mar 16 2022 *)

A172323 a(n) = floor(n*(sqrt(5)+sqrt(2))).

Original entry on oeis.org

0, 3, 7, 10, 14, 18, 21, 25, 29, 32, 36, 40, 43, 47, 51, 54, 58, 62, 65, 69, 73, 76, 80, 83, 87, 91, 94, 98, 102, 105, 109, 113, 116, 120, 124, 127, 131, 135, 138, 142, 146, 149, 153, 156, 160, 164, 167, 171, 175, 178, 182, 186, 189, 193, 197, 200, 204, 208, 211

Offset: 0

Vincenzo Librandi, Feb 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[Floor(n*(Sqrt(5)+Sqrt(2))): n in [0..60]];

With[{c=Sqrt[5]+Sqrt[2]},Floor[c*Range[0,60]]] (* Harvey P. Dale, Sep 03 2012 *)

A046205 In Leibniz's Harmonic Triangle, write numerator first and then denominator of each element.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 6, 1, 3, 1, 4, 1, 12, 1, 12, 1, 4, 1, 5, 1, 20, 1, 30, 1, 20, 1, 5, 1, 6, 1, 30, 1, 60, 1, 60, 1, 30, 1, 6, 1, 7, 1, 42, 1, 105, 1, 140, 1, 105, 1, 42, 1, 7, 1, 8, 1, 56, 1, 168, 1, 280, 1, 280, 1, 168, 1, 56, 1, 8, 1, 9, 1, 72, 1, 252, 1, 504, 1, 630, 1, 504, 1

Offset: 1

Mohammad K. Azarian

			1/1;
1/2, 1/2;
1/3, 1/6, 1/3;
1/4, 1/12, 1/12, 1/4;
1/5, 1/20, 1/30, 1/20, 1/5; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506, A046206, A046212, A046201, A046204, A046208, A007622, A137752, A137753, A137754.

More terms from Gregory D Johnson (gjohn(AT)iname.com)

Edited by M. F. Hasler, Apr 05 2015

A102171 Iccanobirt semiprime indices (1 of 15): Indices of semiprime numbers in A102111.

Original entry on oeis.org

5, 11, 24, 33, 57, 64, 71, 72, 116, 126, 141, 174, 210, 311, 334, 370, 441, 480, 574

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Cf. A001358, A102111, A102172-A102185.

A102111(a(n)) = A102191(n).

a(13) from Robert Price, Nov 07 2018

Offset changed to 1 and a(14)-a(19) from Jinyuan Wang, Jul 31 2021

A128401 Numbers k such that k^2 divides 21^k-1.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 22, 44, 52, 68, 110, 220, 260, 340, 506, 572, 748, 820, 884, 1012, 2530, 2756, 2860, 3740, 3916, 4108, 4420, 5060, 9020, 9724, 10660, 13156, 13780, 13940, 17204, 19580, 20540, 23782, 29084, 30316, 34060, 45188, 46852, 47564

Offset: 1

Alexander Adamchuk, Mar 01 2007

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128402, A128403, A128404.

Select[Range[50000],Divisible[21^#-1,#^2]&] (* Harvey P. Dale, Jun 17 2011 *)

A128403 Numbers k such that k^2 divides 23^k-1.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 12, 20, 22, 24, 40, 42, 44, 60, 66, 78, 84, 88, 120, 132, 156, 168, 212, 220, 264, 312, 420, 424, 440, 444, 462, 474, 546, 620, 636, 660, 780, 820, 840, 858, 888, 924, 948, 1014, 1060, 1092, 1218, 1220, 1240, 1272, 1320, 1560, 1640, 1716

Offset: 1

Alexander Adamchuk, Mar 01 2007

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

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128404.

Join[{1},Select[Range[2000],PowerMod[23,#,#^2]==1&]] (* Harvey P. Dale, May 02 2015 *)

A128405 Numbers k such that k^2 divides 12^k - 1.

Original entry on oeis.org

1, 11, 253, 11891, 768361, 36112967, 61488361, 154261943, 2936791979, 61057324141, 67546215517, 107342336783, 186740152357, 347036920549, 429306186947, 468493520891, 635974117823, 797688507253, 3174672129299

Offset: 1

Alexander Adamchuk, Mar 01 2007

11 divides all except the first term.

All terms are congruent to +-1 (mod 12). - Robert G. Wilson v, Dec 19 2014

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

fQ[n_] := Mod[ PowerMod[12, n, n^2] - 1, n^2] == 0; k = 1; lst = {}; While[k < 100000001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k += 10; If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k += 2]; lst (* Robert G. Wilson v, Dec 19 2014 *)

a(6)-a(8) from Farideh Firoozbakht, Mar 05 2007

Terms a(9) onward from Max Alekseyev, May 06 2010

A130658 Period 4: repeat [1, 1, 2, 2].

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1

Offset: 0

Paul Curtz, Jun 21 2007

Continued fraction expansion of (9+sqrt(221))/14. - Klaus Brockhaus, May 03 2010
From Klaus Brockhaus, May 14 2010: (Start)
Decimal expansion of 34/303.
a(n) = A014695(n+3). (End)
Lengths of runs in A214090. - Reinhard Zumkeller, Jul 06 2012

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

Cf. A014695, A021913, A177156, A214090.

a130658 = (+ 1) . (`div` 2) . (`mod` 4)
a130658_list = cycle [1,1,2,2]  -- Reinhard Zumkeller, Jul 06 2012

&cat [[1, 1, 2, 2]^^30]; // Wesley Ivan Hurt, Jul 11 2016

seq(op([1, 1, 2, 2]), n=0..50); # Wesley Ivan Hurt, Jul 11 2016

PadRight[{}, 100, {1, 1, 2, 2}] (* Wesley Ivan Hurt, Jul 11 2016 *)

a(n)=(n%4>1)+1 \\ Charles R Greathouse IV, Jul 11 2016

G.f.: ( 1+2*x^2 ) / ( (1-x)*(1+x^2) ). - R. J. Mathar, Jan 18 2011
a(n) = (n^3 mod 4 + (n+1)^3 mod 4 + 1)/2. - Gary Detlefs, Apr 15 2011
a(n) = -1/2*cos(1/2*Pi*n)-1/2*sin(1/2*Pi*n)+3/2. - Leonid Bedratyuk, May 13 2012
From Wesley Ivan Hurt, May 30 2015: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n>3.
a(n) = (3+(-1)^((2*n+3+(-1)^n)/4))/2. (End)
From Wesley Ivan Hurt, Jul 11 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = A021913(n) + 1. (End)

More terms from Klaus Brockhaus, May 14 2010

A102165 Iccanobirt primes (15 of 15): Prime numbers in A102125.

Original entry on oeis.org

2, 7, 31, 941, 7112507, 12796921, 3517479344831, 1899587921740207, 57354010293760755391, 35721164922760679029463000239097478253, 7147924589973841766823293744823574255243111

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Next term is too large to include.

Cf. A000040, A102125, A102145, A102151-A102164.

a(n) = A102125(A102145(n)).

Offset changed to 1 by Jinyuan Wang, Aug 14 2021

cp's OEIS Frontend

A128394 Numbers k such that k^2 divides 14^k - 1.

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A128396 Numbers k such that k^2 divides 16^k-1.

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A172323 a(n) = floor(n*(sqrt(5)+sqrt(2))).

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Magma

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A046205 In Leibniz's Harmonic Triangle, write numerator first and then denominator of each element.

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A102171 Iccanobirt semiprime indices (1 of 15): Indices of semiprime numbers in A102111.

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A128401 Numbers k such that k^2 divides 21^k-1.

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A128403 Numbers k such that k^2 divides 23^k-1.

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A128405 Numbers k such that k^2 divides 12^k - 1.

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A130658 Period 4: repeat [1, 1, 2, 2].

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A102165 Iccanobirt primes (15 of 15): Prime numbers in A102125.