A006556 -id:A006556 - cp's OEIS frontend

A006565 Number of ways to color vertices of a hexagon using <= n colors, allowing only rotations.

0, 1, 14, 130, 700, 2635, 7826, 19684, 43800, 88725, 166870, 295526, 498004, 804895, 1255450, 1899080, 2796976, 4023849, 5669790, 7842250, 10668140, 14296051, 18898594, 24674860, 31853000, 40692925, 51489126, 64573614

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Victor Meally, Letter to N. J. A. Sloane, no date.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A027670, A075195.

A006565 := n-> (n^6+n^3+2*n^2+2*n)/6.
A006565:=-(1+7*z+53*z**2+49*z**3+10*z**4)/(z-1)**7; [Conjectured by Simon Plouffe in his 1992 dissertation.]

n*(n+1)*(n^2+n+1)*(n^2-2*n+2)/6.

A211450 (p-1)/x, where p = prime(n) and x = ord(5,p), the smallest positive integer such that 5^x == 1 mod p.

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 1, 2, 1, 2, 10, 1, 2, 1, 1, 1, 2, 2, 3, 14, 1, 2, 1, 2, 1, 4, 1, 1, 4, 1, 3, 2, 1, 2, 4, 2, 1, 3, 1, 1, 2, 12, 10, 1, 1, 6, 6, 1, 1, 2, 1, 2, 6, 10, 1, 1, 4, 10, 1, 2, 1, 1, 1, 2, 39, 1, 2, 3, 1, 2, 1, 2, 3, 1, 18, 1, 4, 1, 16, 24, 2, 2, 2, 1

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211449, A211451, A211452, A211453, A211454, A006556.

nn = 5; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A211449 (p-1)/x, where p = prime(n) and x = ord(4,p), the smallest positive integer such that 4^x == 1 mod p.

Original entry on oeis.org

0, 2, 2, 2, 2, 2, 4, 2, 2, 2, 6, 2, 4, 6, 2, 2, 2, 2, 2, 2, 8, 2, 2, 8, 4, 2, 2, 2, 6, 8, 18, 2, 4, 2, 2, 10, 6, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 6, 2, 6, 8, 2, 20, 10, 32, 2, 2, 2, 6, 8, 6, 2, 6, 2, 4, 2, 22, 16, 2, 2, 8, 2, 2, 2, 2, 2, 2, 18, 4, 4, 2, 2, 10, 12

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211450, A211451, A211452, A211453, A211454, A006556.

nn = 4; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A211451 (p-1)/x, where p = prime(n) and x = ord(6,p), the smallest positive integer such that 6^x == 1 mod p.

Original entry on oeis.org

0, 0, 4, 3, 1, 1, 1, 2, 2, 2, 5, 9, 1, 14, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 8, 10, 1, 1, 1, 1, 1, 1, 1, 6, 4, 1, 1, 6, 2, 4, 1, 3, 10, 2, 14, 1, 2, 1, 1, 1, 1, 14, 12, 1, 1, 2, 2, 1, 1, 5, 2, 2, 6, 62, 6, 2, 2, 6, 1, 3, 11, 2, 1, 1, 6, 2, 4, 1, 1, 24, 1, 15, 10

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211449, A211450, A211452, A211453, A211454, A006556.

nn = 6; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A211452 (p-1)/x, where p = prime(n) and x = ord(7,p), the smallest positive integer such that 7^x == 1 mod p.

Original entry on oeis.org

1, 2, 1, 0, 1, 1, 1, 6, 1, 4, 2, 4, 1, 7, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 4, 8, 1, 2, 2, 2, 2, 1, 3, 1, 2, 1, 1, 15, 19, 8, 2, 2, 1, 6, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 14, 2, 1, 2, 10, 3, 2, 3, 6, 1, 1, 11, 1, 6, 6, 1, 2, 4, 1, 2, 17, 22, 6, 1, 1, 6

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211449, A211450, A211451, A211453, A211454, A006556.

nn = 7; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A211453 (p-1)/x, where p = prime(n) and x = ord(8,p), the smallest positive integer such that 8^x == 1 mod p.

Original entry on oeis.org

0, 1, 1, 6, 1, 3, 2, 3, 2, 1, 6, 3, 2, 3, 2, 1, 1, 3, 3, 2, 24, 6, 1, 8, 6, 1, 6, 1, 9, 4, 18, 1, 2, 3, 1, 30, 3, 3, 2, 1, 1, 3, 2, 6, 1, 6, 3, 6, 1, 3, 8, 2, 30, 5, 16, 2, 1, 6, 3, 4, 3, 1, 9, 2, 6, 1, 33, 48, 1, 3, 4, 2, 6, 3, 3, 2, 1, 9, 2, 6, 1, 3, 10, 18

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211449, A211450, A211451, A211452, A211454, A006556.

nn = 8; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A211454 (p-1)/x, where p = prime(n) and x = ord(9,p), the smallest positive integer such that 9^x == 1 mod p.

Original entry on oeis.org

1, 0, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 10, 2, 2, 2, 2, 12, 6, 2, 12, 2, 2, 2, 4, 2, 6, 2, 4, 2, 2, 2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 4, 2, 24, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 18, 4, 2, 2, 2, 18, 2, 8, 2, 2, 4, 2, 4, 2, 2, 6, 4, 2, 2, 2, 4, 2, 4, 2, 4, 10, 16

Offset: 1

T. D. Noe, Apr 11 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001917, A094593, A211449, A211450, A211451, A211452, A211453, A006556.

nn = 9; Table[If[Mod[nn, p] == 0, 0, (p-1)/MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

A006559 Short period primes: the decimal expansion of 1/p has period less than p-1, but greater than zero.

Original entry on oeis.org

3, 11, 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89, 101, 103, 107, 127, 137, 139, 151, 157, 163, 173, 191, 197, 199, 211, 227, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353, 359, 373, 397, 401, 409, 421, 431, 439, 443, 449, 457

Offset: 1

N. J. A. Sloane

Primes 2 and 5 are excluded because 1/2 and 1/5 have no period. Also primes p whose multiplicative order mod p is less than p-1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Victor Meally, Letter to N. J. A. Sloane, no date.
Index entries for sequences related to decimal expansion of 1/n

Cf. A006883.

Select[Prime[Range[100]], MultiplicativeOrder[10, #] < # - 1 &]

a(n)=gcd(n,10)==1 && isprime(n) && znorder(Mod(10,n))Charles R Greathouse IV, Mar 15 2014

from itertools import islice
from sympy import nextprime, n_order
def A006559_gen(startvalue=1): # generator of terms >= startvalue
    p = max(startvalue-1,1)
    while (p:=nextprime(p)):
        if p!=2 and p!=5 and n_order(10,p)A006559_list = list(islice(A006559_gen(),20)) # Chai Wah Wu, Mar 03 2025

More terms from James Sellers, Aug 21 2000

A060370 Ratios (p-1)/d, where p is a prime and d is the number of digits of the periodic part of the decimal expansion of 1/p.

Original entry on oeis.org

1, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 12, 8, 2, 1, 4, 1, 1, 2, 2, 9, 6, 2, 2, 1, 25, 3, 2, 1, 1, 3, 1, 17, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 7, 1, 2, 1, 1, 34, 8, 5, 1, 1, 1, 54, 4, 10, 2, 2, 2, 2, 1, 4, 3, 1, 2, 3, 11, 2, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 2, 1, 2

Offset: 1

Klaus Brockhaus, Apr 01 2001

The sequence of 2nd, 4th and following terms coincides with A006556, which gives the "number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5".

			a(13) = 40/5 = 8, since 41 is the 13th prime and the periodic part of 1/41 = 0.02439024390... consists of 5 digits.

Carmine Suriano and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 101 terms from Carmine Suriano)

Cf. A000040, A060283, A048595, A006556.

Join[{1, 2, 4}, Table[p = Prime[n]; (p - 1)/Length[RealDigits[1/p, 10][[1, 1]]], {n, 4, 100}]] (* T. D. Noe, Oct 04 2012 *)

from sympy import prime, n_order
def A060370(n): return 1 if n == 1 or n == 3 else n_order(10, prime(n))
print([(prime(n)-1)//A060370(n) for n in range(1,86)]) # Karl-Heinz Hofmann, Mar 16 2022

a(n) = (b(n)-1)/c(n), where b(n) and c(n) are the n-th terms of A000040 and A048595 respectively.

A006596 Numbers k such that (2^(2k+1) - 2^(k+1) + 1)/5 is prime.

Original entry on oeis.org

2, 5, 6, 14, 21, 26, 141, 278, 281, 306, 345, 1365, 2573, 2661, 4766, 5385

Offset: 1

N. J. A. Sloane

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Victor Meally, Letter to N. J. A. Sloane, no date.
S. S. Wagstaff, Jr., The Cunningham Project

For[ i=1, i<=10000, i++, If[ PrimeQ[ ( 2^(2n+1) - 2^(n+1) + 1)/5 ], Print[ n ] ] ]
Select[Range[5400],PrimeQ[(2^(2#+1)-2^(#+1)+1)/5]&] (* Harvey P. Dale, Jun 28 2023 *)

is(n)=ispseudoprime((2^(2*n+1) - 2^(n+1) + 1)/5) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Douglas R. Burke (dburke(AT)nevada.edu)

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A006565 Number of ways to color vertices of a hexagon using <= n colors, allowing only rotations.

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A211450 (p-1)/x, where p = prime(n) and x = ord(5,p), the smallest positive integer such that 5^x == 1 mod p.

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A211449 (p-1)/x, where p = prime(n) and x = ord(4,p), the smallest positive integer such that 4^x == 1 mod p.

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A211451 (p-1)/x, where p = prime(n) and x = ord(6,p), the smallest positive integer such that 6^x == 1 mod p.

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A211452 (p-1)/x, where p = prime(n) and x = ord(7,p), the smallest positive integer such that 7^x == 1 mod p.

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A211453 (p-1)/x, where p = prime(n) and x = ord(8,p), the smallest positive integer such that 8^x == 1 mod p.

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A211454 (p-1)/x, where p = prime(n) and x = ord(9,p), the smallest positive integer such that 9^x == 1 mod p.

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A006559 Short period primes: the decimal expansion of 1/p has period less than p-1, but greater than zero.

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A060370 Ratios (p-1)/d, where p is a prime and d is the number of digits of the periodic part of the decimal expansion of 1/p.

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