A097717 -id:A097717 - cp's OEIS frontend

A291321 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 8.

1012658227848, 1139240506329, 10126582278481012658227848, 11392405063291139240506329, 101265822784810126582278481012658227848, 113924050632911392405063291139240506329, 1012658227848101265822784810126582278481012658227848

Offset: 1

Seiichi Manyama, Aug 22 2017

Let x = (10^12 - 8)/79 = 12658227848. Then a(1) = 8*x*10 + 8, a(2) = 9*x*10 + 9.

For consistency with A146088 (similar for ratio k=2) and others, where an initial a(0) = 0 has been added, the same could be considered here. It would be compatible with the formula given for a(2k). - M. F. Hasler, May 03 2025

			a(1) = b*10 + 8 with b = 101265822784, and 8*a(1) = 8101265822784 = 8*10^12 + b.

Seiichi Manyama, Table of n, a(n) for n = 1..152
Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), A146754 (k=5), A291354 (k=6), A291215 (k=7), this sequence (k=8), A291353 (k=9).
All these are subsequences of A034089 (except for an initial 0 in some of them).
Cf. A092697, A097717.

a(n)=10^((n+1)\2*13)\79*(9-n%2) \\ Charles R Greathouse IV, Aug 23 2017

a(2*k - 1) = 8*(10^(13*k) - 1)/79.

a(2*k) = 9*(10^(13*k) - 1)/79.

Edited by M. F. Hasler, May 03 2025

A291353 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 9.

Original entry on oeis.org

10112359550561797752808988764044943820224719, 1011235955056179775280898876404494382022471910112359550561797752808988764044943820224719

Offset: 1

Seiichi Manyama, Aug 23 2017

For consistency with A146088 (similar for ratio k=2) and others, where an initial a(0) = 0 has been added, the same could be considered here. It would be compatible with the formula given for a(n). - M. F. Hasler, May 03 2025

Seiichi Manyama, Table of n, a(n) for n = 1..22
Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), A146754 (k=5), A291354 (k=6), A291215 (k=7), A291321 (k=8), this sequence (k=9).
All these are subsequences of A034089 (except for an initial 0 in some of them).
Cf. A092697, A097717.

a(n) = 9*(10^(44*n) - 1)/89.

A291354 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 6.

Original entry on oeis.org

1016949152542372881355932203389830508474576271186440677966, 1186440677966101694915254237288135593220338983050847457627, 1355932203389830508474576271186440677966101694915254237288

Offset: 1

Seiichi Manyama, Aug 23 2017

For consistency with A146088 (similar for ratio k=2) and others, where an initial a(0) = 0 has been added, the same could be considered here. It would be compatible with the formula given for a(4k). - M. F. Hasler, May 03 2025

Seiichi Manyama, Table of n, a(n) for n = 1..68
Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), A146754 (k=5), this sequence (k=6), A291215 (k=7), A291321 (k=8), A291353 (k=9).
All these are subsequences of A034089 (except for an initial 0 in some of them).
Cf. A092697, A097717.

a(4*k - 3) = 6*(10^(58*k) - 1)/59.
a(4*k - 2) = 7*(10^(58*k) - 1)/59.
a(4*k - 1) = 8*(10^(58*k) - 1)/59.
a(4*k)     = 9*(10^(58*k) - 1)/59.

A094676 a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end and the second digit of m is not zero.

Original entry on oeis.org

1, 210526315789473684, 3103448275862068965517241379, 410256, 714285, 6101694915254237288135593220338983050847457627118644067796, 7101449275362318840579, 8101265822784, 91011235955056179775280898876404494382022471

Offset: 1

Lekraj Beedassy, Jun 07 2004

Here when the leftmost digit of m is shifted to the right end the number of digits may not decrease - compare A097717.

Least n-transposable number. A k-transposable number, 1 <= k <= 9, is one which is k times the number obtained when the leftmost digit is moved to the end.

			a(4) = 410256 = 4*102564.

H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20; 31-2, Les Editions D'Organisation, Paris 1984.
L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44; 212-3, Dover NY 1959.

Cf. A097717. A249596-A249599.

a(n) = n prepended to n*(10^m - n)/(10*n - 1), where m = A094224(n) - 1.

Edited by N. J. A. Sloane, Apr 13 2009

a(5) corrected by Emilio Martín, Jul 28 2022

A128857 a(n) = least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end.

Original entry on oeis.org

1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966

Offset: 1

Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007

a(n) is simply the decimal period of the fraction n/(10n-1). Thus, we have: n/(10n-1) = a(n)/(10^A128858(n)-1). With the usual convention that the decimal period of 0 is zero, that definition would allow the extension a(0)=0. a(n) is also the period of the decadic integer -n/(10n-1). - Gerard P. Michon, Oct 31 2012

			a(4) = 102564 since this is the smallest number that begins with 1 and which is divided by 4 when the first digit 1 is made the last digit (102564/4 = 25641).

A. V. Chupin, Table of n, a(n) for n=1..101
G. P. Michon, Integers that are divided by k if their digits are rotated left.

Minimal numbers for shifting any digit from the left to the right (not only 1) are in A097717.

By accident, the nine terms of A092697 coincide with the first nine terms of the present sequence. - N. J. A. Sloane, Apr 13 2009

(*Moving digits a:*) Give[a_,n_]:=Block[{d=Ceiling[Log[10,n]],m=(10n-1)/GCD[10n-1, a]}, If[m!=1,While[PowerMod[10,d,m]!=n,d++ ],d=1]; ((10^(d+1)-1) a n)/(10n-1)]; Table[Give[1,n],{n,101}]

from sympy import n_order
def A128857(n): return n*(10**n_order(10,(m:=10*n-1))-1)//m # Chai Wah Wu, Apr 09 2024

Edited by N. J. A. Sloane, Apr 13 2009

Code and b-file corrected by Ray Chandler, Apr 29 2009

A159774 Least number m, written in base n, such that m/2 is obtained merely by shifting the leftmost digit of m to the right end, and 2m by shifting the rightmost digit of m to the left end, digits defined in base n.

Original entry on oeis.org

1012, 102, 102342, 1031345242, 103524563142, 1042, 10467842, 105263157894736842, 316, 10631694842

Offset: 3

William A. Hoffman III (whoff(AT)robill.com), Apr 21 2009

10(b2) and 31(b5) do not both halve and double by rotations. No 2-digit answer can meet the description, so the sequence begins with a base 3 value.

			1042(b8)/2 = 421(b8) and 1042(b8)*2 = 2104(b8)
316 (base 11) = 380 (base 10), 163 (base 11) = 190 (base 10), 631 (base 11) = 760 (base 10).

W. A. Hoffman III, Algorithm to compute terms.

Cf. A092697, A097717, A094224, A094676, A158877.

See A147514 for these numbers written in base 10.

Offset corrected by N. J. A. Sloane, Apr 23 2009
a(11) corrected. To indicate that terms from base n=13 on need digits larger than 9, keywords fini, full added. - Ray Chandler and R. J. Mathar, Apr 23 2009
Edited by Ray Chandler, May 02 2009

cp's OEIS Frontend

A291321 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A291353 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A291354 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A094676 a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end and the second digit of m is not zero.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

Extensions

A128857 a(n) = least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A159774 Least number m, written in base n, such that m/2 is obtained merely by shifting the leftmost digit of m to the right end, and 2m by shifting the rightmost digit of m to the left end, digits defined in base n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions