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

Author

Tanya Khovanova, Jun 13 2021

Keywords

• nonn
• base
• easy
• fini
• full

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

Examples

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

Crossrefs

Cf. A084680, A059892.

Subsequence of A027895.

10th row of A226477.

Programs

Mathematica

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

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

Extensions

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

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