A097717 -id:A097717 - cp's OEIS frontend

A249596 Analog of A097717 in base 2.

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

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

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

A034089 Numbers that are proper divisors of the number you get by rotating digits right once.

Original entry on oeis.org

102564, 128205, 142857, 153846, 179487, 205128, 230769, 102564102564, 128205128205, 142857142857, 153846153846, 179487179487, 205128205128, 230769230769, 1012658227848, 1139240506329, 102564102564102564

Offset: 1

Erich Friedman

Let p(q) denote the period of the fraction q; then sequence is generated by p(i / (10k-1)), k=2,3,4,5,6,7,8,9; k <= i <= 9 and the concatenations of those periods, e.g., p(7/39)=a(5) p(2/19)=a(17).
Example if k=5: p((5+2)/49)=142857 which is in the sequence as the concatenations 142857142857, 142857142857142857, 142857142857142857142857, etc. - Benoit Cloitre, Feb 02 2002
The i in p(i / (10k-1)) is the last digit of the period, while k is equal to the ratio (right-rotated of p)/p. Thus no concatenation of any different such p's can be in the sequence. There are 8*9/2 = 36 terms which are not concatenation of previous terms, the last one being a(124) = 1525423728813559322033898305084745762711864406779661016949 with 58 digits. The term a(3)=p(7/49) is the only period of length (6) different from the length (42) of the other terms corresponding to the same value of k. - M. F. Hasler, Nov 18 2007
Numbers comprising multiple copies of a single digit, e.g., 111111, are not permitted. - Harvey P. Dale, Mar 08 2013
From Emmanuel Vantieghem, Oct 25 2015: (Start)
Subsequence of A245680.
Every element of the sequence is a multiple of 3.
The leading digit of every element is < 5.
(End)

M. F. Hasler, Table of n, a(n) for n = 1..124

Subsequences of this sequence (with quotient k): A146088 (k=2), A146561 (k=3), A146569 (k=4), A146754 (k=5), A291354 (k=6), A291215 (k=7), A291321 (k=8), A291353 (k=9).

Cf. A092697, A097717.

period(p,q,S=[])=until(setsearch(S,p),S=setunion(S,[p]);p=10*p%q);S=[];until(p==S[1],S=concat(S,p);p=10*p%q);S*10\q /* print list of periods, right-rotated and ratio */ rotquo(n,d)={d=divrem(n,10);d[1]+=d[2]*10^#Str(d[1]);[n,d[1],d[1]/n]} for(k=2,9,for(i=k,9,print1( i/(10*k-1),"\t",rotquo(sum(j=1,#p=period(i,k*10-1),p[j]*10^(#p-j))))) /* build the sequence up to the greatest period */ A034089()={local(S=[],p); for(k=2,9,for(i=k,9,S=concat(S,sum(j=1,#p=period(i,k*10-1),p[j]*10^(#p-j))))); S=vecsort(S); for(i=1,#S, for(c=2,58\p=#Str(S[i]), S=concat(S,S[i]*(10^(c*p)-1)/(10^p-1)) )); vecsort(S)} \\ M. F. Hasler, Nov 18 2007

Edited, corrected and extended by M. F. Hasler, Nov 18 2007

A146561 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 3.

Original entry on oeis.org

1034482758620689655172413793, 1379310344827586206896551724, 1724137931034482758620689655, 2068965517241379310344827586, 2413793103448275862068965517, 2758620689655172413793103448, 3103448275862068965517241379, 10344827586206896551724137931034482758620689655172413793

Offset: 1

N. J. A. Sloane, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

For consistency with A146088 (analog for k=2), where an initial a(0) = 0 has been added, the same should be done here. - M. F. Hasler, May 03 2025

Seiichi Manyama, Table of n, a(n) for n = 1..245
Wikipedia, Parasitic number.

Cf. A146088 (k=2), this sequence (k=3), A146569 (k=4), A146754 (k=5), A291354 (k=6), A291215 (k=7), A291321 (k=8), A291353 (k=9).
All these are subsequences of A034089.
Cf. A092697, A097717.

From Seiichi Manyama, Aug 22 2017: (Start)
a(7*k - 6) = 3*(10^(28*k) - 1)/29.
a(7*k - 5) = 4*(10^(28*k) - 1)/29.
a(7*k - 4) = 5*(10^(28*k) - 1)/29.
a(7*k - 3) = 6*(10^(28*k) - 1)/29.
a(7*k - 2) = 7*(10^(28*k) - 1)/29.
a(7*k - 1) = 8*(10^(28*k) - 1)/29.
a(7*k)     = 9*(10^(28*k) - 1)/29. (End)

More terms from Seiichi Manyama, Aug 22 2017

A146754 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 5.

Original entry on oeis.org

142857, 142857142857, 142857142857142857, 142857142857142857142857, 142857142857142857142857142857, 142857142857142857142857142857142857, 102040816326530612244897959183673469387755, 122448979591836734693877551020408163265306, 142857142857142857142857142857142857142857

Offset: 1

N. J. A. Sloane, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

From Seiichi Manyama, Aug 22 2017: (Start)
For k >= 1, (10^(6*k) - 1)/7 is a term.
For 5 <= a <= 9 and k >= 1, a*(10^(42*k) - 1)/49 is a term. (End)
For consistency with A146088 (similar for ratio 2), where an initial a(0) = 0 has been added, the same could be considered here. It would be compatible with the formulas given above (with k = 0). - M. F. Hasler, May 03 2025

			From _Seiichi Manyama_, Aug 22 2017: (Start)
a(1) = b1*10 + 7 with b1 = 14285, and 5*a(1) = 714285 = 7*10^5 + b1.
a(7) = b7*10 + 5 with b7 = 10204081632653061224489795918367346938775, and
  5*a(7) = 510204081632653061224489795918367346938775 = 5*10^41 + b7. (End)

Robert Israel, Table of n, a(n) for n = 1..258
Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), this sequence (k=5), A291354 (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.

f:= proc(d) # solutions with d+1 digits
    local b,R,a;
    R:= NULL;
    for b from ceil(49*10^(d-1)/(10^d - 1)) to 9 do
       a:= (10^d-5)*b/49;
       if a::integer then R:= R, 10*a+b fi
    od;
   R
end proc:
map(f, [$1..42]); # Robert Israel, Nov 05 2024

A146569 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 4.

Original entry on oeis.org

0, 102564, 128205, 153846, 179487, 205128, 230769, 102564102564, 128205128205, 153846153846, 179487179487, 205128205128, 230769230769, 102564102564102564, 128205128205128205, 153846153846153846, 179487179487179487

Offset: 0

N. J. A. Sloane, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

a(13) <= 102564102564102564. - Donovan Johnson, Jun 06 2009

The condition is equivalent to constraining the numbers to be of the form 10*m+d with a k-digit number m and a nonzero digit d such that 4*(10*m+d) = 10^k * d + m, i.e., 39*m = (10^k - 4)*d. Checking modulo 13, this implies k = 5 (mod 6). Also, m >= 10^(k-1) implies d >= 4. Each such k and d leads to a solution. - Hagen von Eitzen, Jun 26 2009

Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), this sequence (k=4), A146754 (k=5), A291354 (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 these).
Cf. A092697, A097717.

a(n) = local(r=(n-1)%6+1,k=(n-r)/6);floor((r+3)/39*10^(6*(k+1))) \\ Hagen von Eitzen, Jun 26 2009

If n = 6*k + r with 1 <= r <= 6, then a(n) = (10^(6*k) - 1)/(10^6 - 1)*a(r) as well as a(n) = floor((r + 3)/39*10^(6*(k+1))). - Hagen von Eitzen, Jun 26 2009

a(7)-a(12) from Donovan Johnson, Jun 06 2009
More terms from Hagen von Eitzen, Jun 26 2009
a(0) = 0 prefixed for consistency with A146088 by M. F. Hasler, May 03 2025

A291215 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 7.

Original entry on oeis.org

1014492753623188405797, 1159420289855072463768, 1304347826086956521739, 10144927536231884057971014492753623188405797, 11594202898550724637681159420289855072463768, 13043478260869565217391304347826086956521739, 101449275362318840579710144927536231884057971014492753623188405797

Offset: 1

Seiichi Manyama, Aug 21 2017

With x = (10^21 - 7)/69 = 14492753623188405797, we have
  a(1) = 7*x*10 + 7, a(2) = 8*x*10 + 8, a(3) = 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(3k). - M. F. Hasler, May 03 2025

			b = 101449275362318840579.
a(1) = b*10 + 7,
7*a(1) = 7101449275362318840579 = 7*10^21 + b.

Robert Israel, Table of n, a(n) for n = 1..135
Wikipedia, Parasitic number.

Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), A146754 (k=5), A291354 (k=6), this (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.

seq(seq(y*((10^(22*k)-1)/69),y=7..9),k=1..6); # Robert Israel, Aug 22 2017

From Robert Israel, Aug 22 2017: (Start)
a(3k-2) = 7(10^(22k)-1)/69.
a(3k-1) = 8(10^(22k)-1)/69.
a(3k) = 9(10^(22k)-1)/69.
a(n+6) = (10^22+1) a(n+3) - 10^22 a(n).
G.f.: x*(1304347826086956521739*x^2 + 1159420289855072463768*x + 1014492753623188405797)/(10^22*x^6 - (10^22+1)*x^3 + 1). (End)

cp's OEIS Frontend

A249596 Analog of A097717 in base 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions

A249599 Analog of A097717 in base 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A249598 Analog of A097717 in base 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A249597 Analog of A097717 in base 3.

Views

Author

Keywords

Links

Crossrefs

Extensions

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A034089 Numbers that are proper divisors of the number you get by rotating digits right once.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A146561 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A146754 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A146569 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 4.

Views

Author

Keywords

Comments