A040402 -id:A040402 - cp's OEIS frontend

A086018 Number of cyclic numbers (A001913) <= 10^n.

0, 1, 9, 60, 467, 3617, 29500, 248881, 2155288, 19016617, 170169241, 1539964486, 14063663530, 129413160100

Offset: 0

Eric W. Weisstein, Jul 07 2003

Note that there are several different definitions of cyclic number: this sequence refers to A001913.

			a(1)=1 since 7 is the only cyclic number <= 10^1.
a(2)=9 since the following are the cyclic numbers <= 10^2: 7, 17, 19, 23, 29, 47, 59, 61, 97.

Eric Weisstein's World of Mathematics, Cyclic Number
Eric Weisstein's World of Mathematics, Full Reptend Prime

Cf. A001913, A040402.

DigitCycleLength[ r_Rational, b_Integer?Positive ] := MultiplicativeOrder[ b, FixedPoint[ Quotient[ #, GCD[ #, b ] ] &, Denominator[ r ] ] ]; a = 0; Do[ If[ Prime[ n ] - DigitCycleLength[ 1/Prime[ n ], 10 ] == 1, a++ ], {n, 2, PrimePi[ 10^7 ]} ] Print[ a ]

Conjectured ratio a(n)/A006880(n) as n->infinity is Artin's constant 0.3739558136...

Extended by Farideh Firoozbakht, Jud McCranie, Ed Pegg Jr and Eric W. Weisstein, Aug 29 2003

a(11)-a(13) from Hiroaki Yamanouchi, Oct 10 2015

A041804 Numerators of continued fraction convergents to sqrt(423).

Original entry on oeis.org

20, 21, 41, 144, 617, 1995, 2612, 4607, 186892, 191499, 378391, 1326672, 5685079, 18381909, 24066988, 42448897, 1722022868, 1764471765, 3486494633, 12223955664, 52382317289, 169370907531, 221753224820, 391124132351, 15866718518860, 16257842651211

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 9214, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A041805, A040402.

Numerator[Convergents[Sqrt[423], 30]] (* Harvey P. Dale, Oct 09 2013 *)

G.f.: -(x^15 -20*x^14 +21*x^13 -41*x^12 +144*x^11 -617*x^10 +1995*x^9 -2612*x^8 -4607*x^7 -2612*x^6 -1995*x^5 -617*x^4 -144*x^3 -41*x^2 -21*x -20) / ((x^8 -96*x^4 +1)*(x^8 +96*x^4 +1)). - Colin Barker, Nov 25 2013

More terms from Colin Barker, Nov 25 2013

A041805 Denominators of continued fraction convergents to sqrt(423).

Original entry on oeis.org

1, 1, 2, 7, 30, 97, 127, 224, 9087, 9311, 18398, 64505, 276418, 893759, 1170177, 2063936, 83727617, 85791553, 169519170, 594349063, 2546915422, 8235095329, 10782010751, 19017106080, 771466253951, 790483360031, 1561949613982, 5476332201977, 23467278421890

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 9214, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A041804, A040402.

I:=[1,1,2,7,30,97,127,224,9087,9311,18398,64505, 276418,893759,1170177,2063936]; [n le 16 select I[n] else 9214*Self(n-8)-Self(n-16): n in [1..50]]; // Vincenzo Librandi, Dec 24 2013

Denominator[Convergents[Sqrt[423], 30]] (* Vincenzo Librandi, Dec 24 2013 *)

G.f.: -(x^14 -x^13 +2*x^12 -7*x^11 +30*x^10 -97*x^9 +127*x^8 -224*x^7 -127*x^6 -97*x^5 -30*x^4 -7*x^3 -2*x^2 -x -1) / ((x^8 -96*x^4 +1)*(x^8 +96*x^4 +1)). - Colin Barker, Nov 25 2013

a(n) = 9214*a(n-8) - a(n-16) for n>15. - Vincenzo Librandi, Dec 24 2013

More terms from Colin Barker, Nov 25 2013

cp's OEIS Frontend

A086018 Number of cyclic numbers (A001913) <= 10^n.

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A041804 Numerators of continued fraction convergents to sqrt(423).

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A041805 Denominators of continued fraction convergents to sqrt(423).

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