A094676 -id:A094676 - cp's OEIS frontend

A097717 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.

1, 105263157894736842, 1034482758620689655172413793, 102564, 714285, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719

Offset: 1

Lekraj Beedassy, Sep 21 2004

			We have a(5)=714285 since 714285/5=142857.
Likewise, a(4)=102564 since this is the smallest number followed by 205128, 307692, 410256, 512820, 615384, 717948, 820512, 923076, ... which all get divided by 4 when the first digit is made last.

R. Sprague, Recreation in Mathematics, Problem 21 pp. 17; 47-8 Dover NY 1963.

A. V. Chupin, Table of n, a(n) for n=1..101
A. V. Chupin, Table of n, a(n) for n=1..154

A097717: when move L digit to R, divides by n (infinite)
A094676: when move L digit to R, divides by n, no. of digits is unchanged (finite)
A092697: when move R digit to L, multiplies by n (finite)
A128857 is the same sequence as A097717 except that m must begin with 1.
Not the same as A092697.
Cf. A249596 - A249599 (bases 2 to 5).

Min[Table[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)], {a,9}]] (* Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007 *)

a(9) from Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007

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

A092697 For 1 <= n <= 9, a(n) = least number m such that the product n*m is obtained merely by shifting the rightmost digit of m to the left end (a finite sequence).

Original entry on oeis.org

1, 105263157894736842, 1034482758620689655172413793, 102564, 142857, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719

Offset: 1

Lekraj Beedassy, Aug 21 2004; corrected Dec 17 2004

This is the least n-parasitic number. A k-parasitic number (where 1 <= k <= 9) is one such that when it is multiplied by k, the product obtained is merely its rightmost digit transferred in front at the leftmost end.

			102564 is 4-parasitic because we have 102564*4=410256.
For n=5: 142857*5=714285. [Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009]

C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford Univ. Press UK 2000.

Wikipedia, Parasitic numbers [From Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009]
P. Yiu, k-left-transposable integers, Chap.18.2 pp. 168/360 in 'Recreational Mathematics'

Cf. A094676, A081463.

For other sequences with the same start, see A128857 and especially the cross-references in A097717.

Edited by N. J. A. Sloane, Apr 13 2009
Corrected to set 5th term to 142857 as this is the least 5-parasitic number. Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009
a(9) added by Ian Duff, Jan 03 2012
Incorrect formula removed by Alois P. Heinz, Feb 18 2020

A249596 Analog of A097717 in base 2.

Original entry on oeis.org

1, 2, 9, 4, 35, 558, 2205, 8, 135, 137970, 33, 1068, 545259, 135926, 138845925, 16, 527, 2106, 35288379945, 2100, 537075, 8382, 2093, 4283544, 1069975, 130, 2294286602622705, 533820, 133371, 146557818382226310, 585910928570692725, 32, 2079

Offset: 1

R. J. Mathar, Mar 30 2009

Conjecture: a(n) = n*A165781(n). - R. J. Mathar, Nov 11 2014

Lars Blomberg, Table of n, a(n) for n = 1..1661

Cf. A094676, A249597, A249598, A249599.

A249596 := proc(n)
    local m,b,mbas,msf ;
    b := 2;
    for m from 1 to 1999999 do
        mbas := convert(m,base,b) ;
        msf := [op(-1,mbas),op(1..nops(mbas)-1,mbas)] ;
        msf := add(op(i,msf)*b^(i-1),i=1..nops(msf)) ;
        if m/n = msf then
            return m;
        end if;
    end do:
    -1 ;
end proc:
for n from 1 do
    print(n,A249596(n)) ;
end do: # R. J. Mathar, Nov 11 2014

a(15)-a(33) from Lars Blomberg, Feb 05 2015

A249599 Analog of A097717 in base 5.

Original entry on oeis.org

1, 16, 3348, 411184, 5, 1262796336, 31415153952, 128, 639, 46402790906782052954848931760, 9548, 37884308119951668432, 507, 483747841655344, 2949712546290578068913640, 368402917173844349535205696, 3162, 1149642179207353109724066230688

Offset: 1

R. J. Mathar, Mar 30 2009

			a(2)=16, because in base 5, 16 is written 31 and 16/2 is 8 and written 13.

Lars Blomberg, Table of n, a(n) for n = 1..369

Cf. A094676, A249596, A249597, A249598.

A249599 := proc(n)
    local m,b,mbas,msf ;
    b := 5;
    for m from 1 to 1999999 do
        mbas := convert(m,base,b) ;
        msf := [op(-1,mbas),op(1..nops(mbas)-1,mbas)] ;
        msf := add(op(i,msf)*b^(i-1),i=1..nops(msf)) ;
        if m/n = msf then
            return m;
        end if;
    end do:
    -1 ;
end proc:
for n from 1 do
    print(n,A249599(n)) ;
end do: # R. J. Mathar, Nov 11 2014

a(6)-a(18) from Lars Blomberg, Feb 05 2015

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

A249598 Analog of A097717 in base 4.

Original entry on oeis.org

1, 18, 279, 4, 68985, 1094166, 49, 264, 1053, 1050, 4191, 17966487875148, 65, 266910, 73278909191113155, 16, 18722068612123127013, 299304917928357795234, 265639, 76514292380672732576340, 1223491190935287357533961, 67880230, 1035, 17360709912, 775

Offset: 1

R. J. Mathar, Mar 30 2009, Nov 11 2014

			279 is (10113)_4 which shift-rotates into (01131)_4 = 93 = 279/3, so 279 qualifies as a(3).

Lars Blomberg, Table of n, a(n) for n = 1..830

Cf. A094676, A249596, A249598, A249599.

a(12)-a(25) from Lars Blomberg, Feb 04 2015

A249597 Analog of A097717 in base 3.

Original entry on oeis.org

1, 32, 3, 88, 260, 15192960, 28, 61616, 9, 7888549122400, 55, 182208, 132538588, 2240, 165, 32048741728, 1185506696, 2194329698227926780769440, 247, 23264534699960, 69737318935284, 179872, 14559920, 16912071582760464, 130885300, 69521680967024, 27

Offset: 1

R. J. Mathar, Mar 30 2009

Lars Blomberg, Table of n, a(n) for n = 1..709

Cf. A094676, A249596, A249598, A249599.

a(6)-a(26) from Lars Blomberg, Feb 04 2015

cp's OEIS Frontend

A097717 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.

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A092697 For 1 <= n <= 9, a(n) = least number m such that the product n*m is obtained merely by shifting the rightmost digit of m to the left end (a finite sequence).

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A249596 Analog of A097717 in base 2.

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A249599 Analog of A097717 in base 5.

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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.

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A249598 Analog of A097717 in base 4.

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A249597 Analog of A097717 in base 3.

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