A345319 Numbers whose reciprocals have period 10.

451, 1353, 2981, 4059, 8943, 9091, 26829, 27273, 81819, 100001, 122221, 300003, 366663, 372731, 900009, 1099989, 1118193, 2463661, 3354579, 4100041, 7390983, 12300123, 22172949, 27100271, 36900369, 81300813, 101010101, 243902439, 303030303, 909090909, 1111111111, 3333333333, 9999999999

Offset

1,1

Comments

Equivalently, these are numbers k such that the multiplicative order of 10 modulo k is 10.

These are indices of terms at which 10 appears in A084680.

There are exactly A059892(10) = mu(10/10)*d(10^10-1) + mu(10/5)*d(10^5-1) + mu(10/2)*d(10^2-1) + mu(10/1)*d(10^1-1) = 48 - 12 - 6 + 3 = 33 terms, where d = A000005 and mu = A008683. - Jianing Song, Jun 15 2021

Links

Table of n, a(n) for n=1..33.

Example

1/451 = 0.00221729490022172949002217294900..., whose periodic part is 0022172949.

Mathematica

Select[Range[100000000], MultiplicativeOrder[10, #] == 10 &]

Prog

(PARI) isok(k) = gcd(k, 10) && (znorder(Mod(10, k)) == 10); \\ Michel Marcus, Jun 14 2021
(PARI) my(v=divisors(10^10-1)); select(x->(znorder(Mod(10, x))==10), v) \\ Jianing Song, Jun 15 2021

Crossrefs

Cf. A084680, A059892.

Subsequence of A027895.

10th row of A226477.

Sequence in context: A145492 A269763 A020268 * A066322 A128922 A224560

Adjacent sequences: A345316 A345317 A345318 * A345320 A345321 A345322

Keyword

nonn,base,easy,fini,full

Author

Tanya Khovanova, Jun 13 2021

Extensions

a(27)-a(28) from Jinyuan Wang, Jun 13 2021

a(29)-a(33) from Jianing Song, Jun 15 2021

Status

approved