A010874 -id:A010874 - cp's OEIS frontend

A032796 Numbers that are congruent to {1, 2, 3, 5, 6} mod 7.

1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99

Offset: 1

Patrick De Geest, May 15 1998

If k is a term, then k*(k+1)*(k+2)*...*(k+6)/(k+(k+1)+(k+2)+...+(k+6)) is a multiple of k.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A032765..A032798.

Cf. A010874.

[n: n in [0..120] | n mod 7 in {1, 2, 3, 5, 6}]; // Vincenzo Librandi, Dec 29 2010

#+{1,2,3,5,6}&/@(7*Range[0,15])//Flatten (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{1,2,3,5,6,8},100] (* Harvey P. Dale, Oct 07 2018 *)

Equals natural numbers minus '4, 7, 11, 14, 18, ...' (= previous term +3, +4, +3, +4, ...).
G.f.: x*(x^5 + x^4 + 2*x^3 + x^2 + x + 1)/((1-x)*(1-x^5)).
a(n) = (m^3 - 6*m^2 + 17*m + 6*(7*floor(n/5)-1))/6, where m = n mod 5. - Luce ETIENNE,Oct 17 2018

A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 175, 186, 198, 211, 225, 240, 240, 241, 243, 246, 250, 255, 261, 268, 276, 285, 295, 306, 318, 331, 345, 360, 360, 361, 363, 366, 370, 375, 381, 388

Offset: 0

Hieronymus Fischer, Jun 11 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A130486, A010872, A010873, A010874, A010875, A010876, A010878, A130481, A130482, A130483, A130484, A130485, A130487.

Accumulate[Mod[Range[0,60],16]] (* Harvey P. Dale, May 30 2020 *)

a(n)=120*floor(n/16)+A130909(n)*(A130909(n)+1)/2. - G.f.: g(x)=(sum{1<=k<16, k*x^k})/((1-x^16)(1-x)). Also: g(x)=x(15x^16-16x^15+1)/((1-x^16)(1-x)^3).

a(n) = +a(n-1) +a(n-16) -a(n-17). G.f. ( x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +14*x^13 +15*x^14) ) / ( (1+x) *(1+x^2) *(1+x^4) *(1+x^8) *(x-1)^2 ). - R. J. Mathar, Nov 05 2011

A261872 a(n) = phi(n) mod 5, where phi is the Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 1, 4, 1, 4, 0, 4, 2, 1, 3, 3, 1, 1, 3, 3, 2, 0, 2, 3, 0, 2, 3, 2, 3, 3, 0, 1, 0, 1, 4, 2, 1, 3, 4, 1, 0, 2, 2, 0, 4, 2, 1, 1, 2, 0, 2, 4, 2, 3, 0, 4, 1, 3, 3, 1, 0, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 4, 2, 1, 0, 1, 0, 4, 3, 2, 4, 0, 2, 4, 4, 2, 1, 0, 3, 4, 2, 4, 0, 1, 2, 2, 1, 2, 0, 0, 0, 2, 2, 3, 3

Offset: 1

Vincenzo Librandi, Sep 04 2015

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A010874, A074942, A084300.

Magma
```
[EulerPhi(n) mod 5: n in [1..110]];
```
Mathematica
```
Table[Mod[EulerPhi[n], 5], {n, 110}]
```

a(n) = eulerphi(n) % 5; \\ Michel Marcus, Sep 05 2015

a(n) = A000010(n) mod 5 = A010874(A000010(n)).

More terms from Antti Karttunen, Dec 04 2017

A349767 Numbers m such that 2^m - m is divisible by 5.

Original entry on oeis.org

3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57, 63, 74, 76, 77, 83, 94, 96, 97, 103, 114, 116, 117, 123, 134, 136, 137, 143, 154, 156, 157, 163, 174, 176, 177, 183, 194, 196, 197, 203, 214, 216, 217, 223, 234, 236, 237, 243, 254, 256, 257, 263, 274, 276, 277, 283, 294, 296, 297, 303

Offset: 1

Bernard Schott, Dec 10 2021

nonn

For every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 5.

Equivalently, numbers that are congruent to {3, 14, 16, 17, 23, 34, 36, 37, 43, 54, 56, 57} mod 60, <==> numbers that are congruent to {+-3, +-14, +-16, +-17, +-23, +-34} mod 60.

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
Index to sequences related to Olympiads.

Cf. A000325, A070402, A010874.

Similar with: A299174 (p = 2), A047257 (p = 3), this sequence (p = 5).

filter:= n -> 2^n-n mod 5 = 0 : select(filter, [$1..400]);

Select[Range[300], PowerMod[2, #, 5] == Mod[#, 5] &] (* Amiram Eldar, Dec 10 2021 *)

isok(m) = Mod(2, 5)^m == Mod(m, 5); \\ Michel Marcus, Dec 10 2021

def ok(n): return pow(2, n, 5) == n%5
print([k for k in range(357) if ok(k)]) # Michael S. Branicky, Dec 10 2021

A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4

Offset: 0

N. J. A. Sloane

Partial sums are given by A130485(n)+n+1. - Hieronymus Fischer, Jun 08 2007

Decimal expansion of 1234567/9999999 = 0.123456712345671234567... - Eric Desbiaux, Nov 03 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A002264, A002265, A002266, A004526, A010872, A010873, A010874, A010875, A010876, A010877, A130485.

Cf. A177160 (decimal expansion of (4502+sqrt(29964677))/6961).

&cat [[1, 2, 3, 4, 5, 6, 7]^^20]; // Wesley Ivan Hurt, Jul 18 2016

seq(op([1, 2, 3, 4, 5, 6, 7]), n=0..20); # Wesley Ivan Hurt, Jul 18 2016

PadRight[{}, 100, {1, 2, 3, 4, 5, 6, 7}] (* Wesley Ivan Hurt, Jul 18 2016 *)

a(n)=n%7+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 7). - Paolo P. Lava, Nov 21 2006
a(n) = A010876(n) + 1. G.f.: (Sum_{k=0..6} (k+1)*x^k)/(1-x^7). Also (7*x^8-8*x^7+1)/((1-x^7)*(1-x)^2). - Hieronymus Fischer, Jun 08 2007
From Wesley Ivan Hurt, Jul 18 2016: (Start)
a(n) = a(n-7) for n>6.
a(n) = 1 - 6*floor(n/7) + Sum_{k=1..6} floor((n + k)/7). (End)

A014064 Coefficients of the reciprocal of the 55th cyclotomic polynomial.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Simon Plouffe

Periodic with period length 55. - Ray Chandler, Apr 03 2017

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

Cf. similar sequences listed in A240328.

Cf. A010880, A010874.

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);

CoefficientList[Series[1/Cyclotomic[55, x], {x, 0, 200}], x] (* Vincenzo Librandi, Apr 05 2014 *)
LinearRecurrence[{1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1},{1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -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},81] (* Ray Chandler, Sep 15 2015 *)

Vec(1/polcyclo(55) + O(x^99)) \\ Michel Marcus, Jan 25 2019

a(n) = (5*w - 1)*(w - 2)*(w - 3)*(w - 4)*(63*m^4 - 350*m^3 + 630*m^2 - 325*m + 12)*(m - 5)!/(43545600*(m - 11)!), where m = n mod 11 and w = floor(n/11) mod 5. - Luce ETIENNE, Nov 21 2018

Name edited by Wolfdieter Lang, Jan 25 2019

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A135033 Period 5: repeat [2, 4, 6, 8, 0].

Original entry on oeis.org

2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0

Offset: 0

William A. Tedeschi, Feb 10 2008

"Cheerleader" sequence: reminiscent of the old cheer, "Two, Four, Six, Eight, Who do we appreciate?"

			a(0) = 2*((0+1) mod 5) = 2*(1 mod 5) = 2.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A010874.

PadRight[{},120,{2,4,6,8,0}] (* Harvey P. Dale, Nov 21 2018 *)

a(n) = 2*((n+1) mod 5).
From R. J. Mathar, Feb 14 2008: (Start)
a(n) = 2*A010874(n+1).
O.g.f.: -2*(1 + 2x + 3x^2 + 4x^3)/((x-1)*(1 + x + x^2 + x^3 + x^4)). (End)

A156551 Period 10: repeat [8,6,0,4,2,2,4,0,6,8].

Original entry on oeis.org

8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8, 8, 6, 0, 4, 2, 2, 4, 0, 6, 8

Offset: 0

Paul Curtz, Feb 09 2009

See A131715.

Cf. A010874, A131715, A155110.

a(n) = A155110(n) mod 10.
G.f.: 2*(4*x^8-x^7+x^6+x^5+x^3+x^2-x+4)/((1-x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)). - R. J. Mathar, Feb 23 2009
a(n) = (15*(4-m^4+8*m^3-19*m^2+12*m)-2*(m^3-6*m^2+17*m-18)*(-1)^floor(n/5))/12 where m = n-5*floor(n/5). - Luce ETIENNE, Oct 13 2017

Edited by R. J. Mathar, Feb 23 2009

More terms from Jinyuan Wang, Feb 26 2020

A255680 a(n) = n(n mod 3)(n mod 5).

Original entry on oeis.org

0, 1, 8, 0, 16, 0, 0, 14, 48, 0, 0, 22, 0, 39, 112, 0, 16, 68, 0, 76, 0, 0, 44, 138, 0, 0, 52, 0, 84, 232, 0, 31, 128, 0, 136, 0, 0, 74, 228, 0, 0, 82, 0, 129, 352, 0, 46, 188, 0, 196, 0, 0, 104, 318, 0, 0, 112, 0, 174, 472, 0, 61, 248, 0, 256, 0, 0, 134, 408, 0, 0, 142, 0, 219, 592, 0, 76, 308, 0, 316, 0, 0, 164, 498, 0, 0, 172, 0

Offset: 0

Zak Seidov, Mar 01 2015

a(n) = 0 for n = 3k and 5k, k=0,1,2,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, -1).

Cf. A255642.

[n*(n mod 3)*(n mod 5): n in [0..80]]; // Vincenzo Librandi, Mar 03 2015

Table[x*Mod[x, 3]*Mod[x, 5], {x, 0, 100}]
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1},{0,1,8,0,16,0,0,14,48,0,0,22,0,39,112,0,16,68,0,76,0,0,44,138,0,0,52,0,84,232},100] (* Harvey P. Dale, Sep 06 2015 *)

vector(101, n, (n-1)*((n-1)%3)*((n-1)%5))

Empirical g.f.: x*(8*x^28 + 6*x^27 + 8*x^25 + 42*x^22 + 16*x^21 + 44*x^18 + 52*x^16 + 14*x^15 + 112*x^13 + 39*x^12 + 22*x^10 + 48*x^7 + 14*x^6 + 16*x^3 + 8*x + 1) / ((x - 1)^2*(x^2 + x + 1)^2*(x^4 + x^3 + x^2 + x + 1)^2*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)^2). - Colin Barker, Mar 02 2015

a(n) = n*A010872(n)*A010874(n). - Michel Marcus, Mar 03 2015

Typo in first Mathematica program corrected by Harvey P. Dale, Jul 03 2021

cp's OEIS Frontend

A032796 Numbers that are congruent to {1, 2, 3, 5, 6} mod 7.

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A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

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A261872 a(n) = phi(n) mod 5, where phi is the Euler totient function.

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A349767 Numbers m such that 2^m - m is divisible by 5.

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A010886 Period 7: repeat [1, 2, 3, 4, 5, 6, 7].

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A014064 Coefficients of the reciprocal of the 55th cyclotomic polynomial.

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Maple

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A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

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A255680 a(n) = n(n mod 3)(n mod 5).