A010874 -id:A010874 - cp's OEIS frontend

A257164 Period 5 sequence: repeat [0, 2, 4, 1, 3].

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

Offset: 0

Wesley Ivan Hurt, Apr 16 2015

Label the vertices of a regular pentagon from 0..4 going clockwise. Then, starting at vertex "0", a(n) gives the order in which the vertices must be connected to draw a clockwise inscribed, 5-pointed star that remains unbroken during construction.

			0 -> 2 -> 4 -> 1 -> 3 -> ..repeat

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

Cf. A005843.

Bisection of A010874.

Magma
```
[(2*n mod 5) : n in [0..100]];
```

A257164:=n->(2*n mod 5): seq(A257164(n), n=0..100);

Mod[2 Range[0, 100], 5] (* or *)
CoefficientList[Series[x (2 + 4 x + x^2 + 3 x^3)/(1 - x^5), {x, 0, 100}], x]
LinearRecurrence[{0, 0, 0, 0, 1}, {0, 2, 4, 1, 3}, 105] (* or *)
NestList[# /. {0 -> 2, 1 -> 3, 2 -> 4, 3 -> 0, 4 -> 1} &, {0}, 104] // Flatten (* Robert G. Wilson v, Apr 30 2015 *)

a(n)=2*n%5 \\ Charles R Greathouse IV, Apr 21 2015

a(n) = (2n mod 5) = A010874(A005843(n)).
G.f.: x*(2+4*x+x^2+3*x^3)/(1-x^5).
Recurrence: a(n) = a(n-5).
a(n) = a(a(a(a(a(n))))).
a(-n) = A010874(3n) = a(a(a(n))).
Bisections: a(2n) = A010874(-n) = a(a(n)); a(2n+1) = A010874(2-n).
Trisections: a(3n) = A010874(n) = a(a(a(a(n)))); a(3n+1) = A010874(n+2); a(3n+2) = A010874(n-1).

A308776 Counterclockwise square spiral of distinct positive integers constructed by greedy algorithm, such that for k = 1..5, every term equal to k mod 5 (except k itself) is vertically or horizontally adjacent to a cell already holding a value equal to k mod 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 16, 7, 12, 17, 8, 13, 18, 9, 10, 15, 20, 21, 26, 31, 36, 41, 22, 27, 32, 37, 42, 23, 28, 33, 38, 43, 14, 19, 24, 29, 25, 30, 35, 40, 45, 50, 46, 51, 56, 47, 52, 57, 62, 67, 72, 77, 48, 53, 58, 63, 68, 73, 78, 34, 39, 44, 49, 54, 59, 55

Offset: 1

Rémy Sigrist, Jun 24 2019

nonn

For k = 1..5, let M_k be the set of lattice points with a value equal to k mod 5; the sets M_1, ..., M_5 form 5 arms that spiral around the origin (see representation in Links section). As a consequence, the sequence alternates runs of terms in arithmetic progression with first difference 5.
This sequence appears to be a permutation of the natural numbers.
This sequence has similarities with A308505.

			The spiral begins:
   89---84---79---74---69---64--123--118--113--108--103
    |                                                 |
   94   54---49---44---39---34---78---73---68---63   98
    |    |                                       |    |
   99   59   29---24---19---14---43---38---33   58   93
    |    |    |                             |    |    |
   95   55   25   15---10----9---18---13   28   53   88
    |    |    |    |                   |    |    |    |
  100   60   30   20    5----4----3    8   23   48   83
    |    |    |    |    |         |    |    |    |    |
  105   65   35   21    6    1----2   17   42   77  112
    |    |    |    |    |              |    |    |    |
  110   70   40   26   11---16----7---12   37   72  107
    |    |    |    |                        |    |    |
  115   75   45   31---36---41---22---27---32   67  102
    |    |    |                                  |    |
  120   80   50---46---51---56---47---52---57---62   97
    |    |                                            |
  125   85---90---61---66---71---76---81---82---87---92
    |
  130--135--140---86---91---96--101--106--111--116--117

Rémy Sigrist, Table of n, a(n) for n = 1..10201
Rémy Sigrist, Colored representation of the spiral for -500 <= x <= 500 and -500 <= y <= 500 (where the color is function of a(n) mod 5)
Rémy Sigrist, Colored scatterplot of (n, a(n)-n) for n = 1..40401 (where the color is function of a(n) mod 5)
Rémy Sigrist, PARI program for A308776

Cf. A010874, A308505.

PARI
```
See Links section.
```

A372808 a(n) = sum of the digits (mod 5) of 5^n.

Original entry on oeis.org

1, 0, 2, 3, 3, 6, 4, 8, 10, 11, 10, 18, 18, 13, 9, 14, 18, 26, 24, 29, 26, 27, 27, 29, 32, 37, 34, 34, 40, 38, 36, 39, 46, 49, 38, 47, 39, 49, 44, 54, 60, 57, 60, 64, 66, 71, 52, 48, 55, 63, 71, 67, 70, 59, 52, 52, 71, 85, 96, 96, 84, 89, 87, 85, 76, 74, 71, 80

Offset: 0

Paolo Xausa, May 13 2024

			a(7) = 8 since 5^7 = 78125 and (7 mod 5) + (8 mod 5) + (1 mod 5) + (2 mod 5) + (5 mod 5) = 2 + 3 + 1 + 2 + 0 = 8.

Paolo Xausa, Table of n, a(n) for n = 0..10000
J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.

Cf. A000351, A010874, A066001.

Array[Total[Mod[IntegerDigits[5^#], 5]] &, 100, 0]

a(n) = my(d=digits(5^n)); vecsum(apply(x->(x % 5), d)); \\ Michel Marcus, May 17 2024

Sum_{n >= 1} a(n)/5^n = 1/9. See Example 5.1 (e) in Borwein and Borwein (1992), p. 639.

A374015 Residue modulo 5 of n! divided by the highest power of 10 which divides n!.

Original entry on oeis.org

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

Offset: 0

Cezary Glowacz, Jun 25 2024

The sequence is not eventually periodic. This because by induction on k the eventual period must be a multiple of 5^k for every k.
a(5^k) = 2^k mod 5.
From Cezary Glowacz, Feb 07 2025: (Start)
The proportions p(m,s) of counts of pairs of consecutive terms s among a(1),...,a(m) converge to equidistribution (and as an immediate consequence, so do proportions of individual terms).
This can be seen, for example, by stating p(5^(4(n+1)+1)-1,s) as affine functions of p(5^(4n+1)-1,t) and examining the convergence of p(5^(4n+1)-1,u) to the equidistribution. Then, p(m,s) converges to the equidistribution because the maximum over s of the absolute values of deviations from 1/16 of p(m,s) for m>k*5^(4n+1)-1 is less than the corresponding maximum over t for p(5^(4n+1)-1,t) plus 2/(5^(4n+1)) + 1/k.
Consecutive terms 1,2,3 do not occur, so that triples do not have a similar equidistribution.
(End)
If n > 0 is not divisible by 5, a(n) == n * a(n-1) (mod 5). - Robert Israel, Jul 05 2024

			a(5) = 1*2*3*4*5/10 mod 5 = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A004154, A010874, A145679.

Cf. A045549, A045547, A045550, A045548.

a:= n-> (f-> irem(f/10^padic[ordp](f, 10), 5))(n!):
seq(a(n), n=0..105);  # Alois P. Heinz, Jun 25 2024

a[n_]:=Mod[n!/10^IntegerExponent[n!, 10],5]; Array[a,106,0] (* Stefano Spezia, Jun 25 2024 *)

a(n)=if(n>4, my(k=n\5); return(lift((n%5)!*a(k)*Mod(2,5)^k))); n!%5 \\ Charles R Greathouse IV, Jan 24 2025

v=[[((1,1,2,1,4)[j]*2**(i*j))%5 for j in range(5)] for i in range(4)]
def a(n):
    c,p=0,1
    while n: c,n,p=(c+1)%4,n//5,(v[c][n%5]*p)%5
    return(p) # Cezary Glowacz, Feb 05 2025

a(n) = A010874(A004154(n)).

A126045 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 5.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A010874, A126043-A126059.

Array[Mod[MersennePrimeExponent@ #, 5] &, 45] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = A010874(A000043(n)). - Michel Marcus, Apr 07 2018

a(45)-a(47) from Ivan Panchenko, Apr 08 2018

a(48) from Amiram Eldar, Oct 14 2024

A137936 a(n) = 5*mod(n,5) + floor(n/5).

Original entry on oeis.org

0, 5, 10, 15, 20, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 5, 10, 15, 20, 25, 6, 11, 16, 21, 26, 7, 12, 17, 22, 27, 8, 13, 18, 23, 28, 9, 14, 19, 24, 29, 10, 15, 20, 25, 30, 11, 16, 21, 26, 31, 12, 17, 22, 27, 32, 13, 18, 23, 28, 33, 14, 19, 24, 29, 34

Offset: 0

William A. Tedeschi, Mar 06 2008

			a(0) = 5*mod(0,5) + floor(0/5) = 0
a(3) = 5*mod(3,5) + floor(3/5) = 15

Cf. A010874, A002266.

Python
```
a = lambda n: 5*(n%5) + floor(n/5)
```

a(n) = 5*mod(n,5) + floor(n/5) = 5*A010874(n) + A002266(n)

O.g.f.: -x(-5x^3+19x^4-5x^2-5x-5)/[(-1+x)^2*(x^3+x^4+x^2+x+1)] . - R. J. Mathar, Mar 07 2008

A358012 Minimal number of coins needed to pay n cents using coins of denominations 1 and 5 cents.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 12, 13, 14, 15, 16, 13, 14, 15, 16, 17, 14, 15, 16, 17, 18, 15, 16

Offset: 0

Sandra Snan, Oct 24 2022

Sequence consists of runs of five consecutive integers: 0..4, 1..5, 2..6, 3..7, etc.

Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A002266, A010874.

Cf. A076314 (1,10 cents), A053344 (1,5,10,25 cents).

Array[Total@ QuotientRemainder[#, 5] &, 77, 0] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = n\5 + n%5 \\ Thomas Scheuerle, Oct 24 2022

a(n) = vecsum(divrem(n, 5)); \\ Michel Marcus, Nov 03 2022

def a(n): return n//5 + n%5 # Michael S. Branicky, Nov 03 2022

Sum of quotient and remainder of n/5.

a(n) = A002266(n) + A010874(n).

cp's OEIS Frontend

A257164 Period 5 sequence: repeat [0, 2, 4, 1, 3].

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A308776 Counterclockwise square spiral of distinct positive integers constructed by greedy algorithm, such that for k = 1..5, every term equal to k mod 5 (except k itself) is vertically or horizontally adjacent to a cell already holding a value equal to k mod 5.

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PARI

A372808 a(n) = sum of the digits (mod 5) of 5^n.

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Mathematica

PARI

Formula

A374015 Residue modulo 5 of n! divided by the highest power of 10 which divides n!.

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Maple

Mathematica

PARI

Python

Formula

A126045 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 5.

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A137936 a(n) = 5*mod(n,5) + floor(n/5).

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Formula

A358012 Minimal number of coins needed to pay n cents using coins of denominations 1 and 5 cents.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python