A087001 -id:A087001

Search: a087001 -id:a087001

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A087000

Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

+10
5

3, 1, 3, 8, 9, 11, 14, 23, 29, 30, 4, 22, 48, 2, 17, 54, 56, 21, 65, 4, 23, 74, 39, 83, 89, 90, 96, 49, 15, 111, 114, 116, 15, 25, 128, 131, 134, 14, 73, 156, 55, 168, 58, 16, 183, 93, 189, 191, 194, 100, 102, 209, 70, 216, 16, 76, 230, 77, 243, 245, 249, 251

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) appears to be the least k such that 10^k+1 is divisible by A028416(n). See A001271. - Michel Marcus, Aug 13 2023

LINKS

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

Index entries for sequences related to decimal expansion of 1/n.

FORMULA

a(n) = A002371(A049084(A028416(n)))/2.

a(n) = A055642(A086999(n))/2.

a(n) = A055642(A087001(n)) = A055642(A087002(n)).

CROSSREFS

Cf. A028416, A002371, A086999, A087001, A087002, A001271, A062397 (10^n+1).

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Jul 29 2003

STATUS

approved

A087002

Right half of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

+10
4

857, 0, 230, 41176470, 473684210, 56521739130, 65517241379310, 78723404255319148936170, 83050847457627118644067796610, 836065573770491803278688524590, 8630, 8876404494382022471910

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) = A086999(n) mod 10^A087000(n); A055642(a(n))=A087000(n);

A087001(n) + a(n) = 10^A087000(n) - 1.

REFERENCES

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, Die periodischen Dezimalbrueche.

LINKS

Table of n, a(n) for n=1..12.

Index entries for sequences related to decimal expansion of 1/n.

Eric Weisstein's World of Mathematics, Midy's Theorem

EXAMPLE

p=19: A086999(5)=526315789473684210 -> [526315789][473684210] ->

A087001(5)=526315789, A087002(5)=473684210,

A087001(5)+A087002(5)=526315789+473684210=999999999.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Jul 29 2003

STATUS

approved

A086999

Periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

+10
3

142857, 90, 769230, 5882352941176470, 526315789473684210, 4347826086956521739130, 3448275862068965517241379310, 2127659574468085106382978723404255319148936170

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A087001(n)=floor(a(n)/10^A087000(n)), A087002(n)=a(n) mod 10^A087000(n);

A087001(n) + A087002(n) = 10^A087000(n) - 1;

a(n) = A087001(n)*10^A087000(n) + A087002(n).

REFERENCES

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, Die periodischen Dezimalbrueche.

LINKS

Table of n, a(n) for n=1..8.

Eric Weisstein's World of Mathematics, Decimal Expansion

Eric Weisstein's World of Mathematics, Repeating Decimal

Eric Weisstein's World of Mathematics, Midy's Theorem

Index entries for sequences related to decimal expansion of 1/n.

EXAMPLE

p=73: a(11)=A060283(21)=13698630 -> [1369][8630] ->

A087001(11)=1369, A087002(11)=8630, A087001(11)+A087002(11)=1369+8630=9999.

CROSSREFS

a(n)=A060283(A049084(A0A028416(n))), A002283.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Jul 29 2003

STATUS

approved

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