cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

Original entry on oeis.org

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

Views

Author

Lekraj Beedassy, Sep 21 2004

Keywords

Examples

			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.

References

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Extensions

a(9) from Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007
Code and b-file corrected by Ray Chandler, Apr 29 2009

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

Views

Author

Lekraj Beedassy, Jun 07 2004

Keywords

Comments

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.

Examples

			a(4) = 410256 = 4*102564.

References

  • 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.

Crossrefs

Cf. A097717. A249596-A249599.

Formula

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

Extensions

Edited by N. J. A. Sloane, Apr 13 2009
a(5) corrected by Emilio Martín, Jul 28 2022

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

Views

Author

R. J. Mathar, Mar 30 2009

Keywords

Comments

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

Links

Crossrefs

Cf. A094676, A249597, A249598, A249599.

Programs

  • Maple
    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

Extensions

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

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

Cf. A094676, A249596, A249598, A249599.

Extensions

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

Views

Author

R. J. Mathar, Mar 30 2009

Keywords

Links

Crossrefs

Cf. A094676, A249596, A249598, A249599.

Extensions

a(6)-a(26) from Lars Blomberg, Feb 04 2015
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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