A146088 -id:A146088 - cp's OEIS frontend

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

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)

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

Original entry on oeis.org

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.

A217592 Integers that are cut in half by cycling all the decimal digits one place to the left.

Original entry on oeis.org

0, 105263157894736842, 210526315789473684, 315789473684210526, 421052631578947368, 526315789473684210, 631578947368421052, 736842105263157894, 842105263157894736, 947368421052631578, 105263157894736842105263157894736842, 210526315789473684210526315789473684

Offset: 0

Gerard P. Michon, Oct 28 2012

A rotation of the decimal digits to the left puts the leading digit in the rightmost position (123456 becomes 234561). The (even) integers that are cut in half by this operation form 10 families; a singleton {a(0)=0} and 9 infinite families, each consisting of any number of repetitions of an 18-digit pattern.
Each of those families corresponds to a single decadic solution of the equation x=2(10x+d) where d is a digit from 0 to 9 (namely x=-2d/19 which is a periodic decadic).
This sequence is strongly related to A146088:
A146088(8k+1) = a(9k+2)/2 = a(9k+1)
A146088(8k+2) = a(9k+3)/2
A146088(8k+3) = a(9k+4)/2 = a(9k+2)
A146088(8k+4) = a(9k+5)/2
A146088(8k+5) = a(9k+6)/2 = a(9k+3)
A146088(8k+6) = a(9k+7)/2
A146088(8k+7) = a(9k+8)/2 = a(9k+4)
A146088(8k+8) = a(9k+9)/2
Note that a(9k+1)/2 is not part of A146088, because rotating one place to the left a decimal expansion starting with 105263157894736842 yields a discarded zero leading digit which is not retrieved by rotating the digits to the right (right-rotation would double the pattern 052631578947368421 but not the number 52631578947368421).
Lists normally have offset 1, but there is a good reason to make an exception in this case. - N. J. A. Sloane, Dec 24 2012

			a(0) = 0 because rotating a lone zero gives 0 = 0/2.
a(1) = 105263157894736842 because its half equals 052631578947368421 (discarding leading 0).
a(3) = 315789473684210526 since a(3)/2 = 157894736842105263.

G. P. Michon, Numbers cut in half by rotating digits left (2006).

A146088(8k+i)=a(9k+i+1)/2 for i=1..8 and any k.

a(j)=j*105263157894736842 for j = 0 to 9
a(n+9)=10^18*a(n) + a(n) mod 10^18 for n>0
a(9k-9+j) = j*a(1)*(10^18k-1)/(10^18-1) for j=1..9 and k>0
e.g., a(1000) = a(9*112-9+1) = a(1)*(10^2016-1)/(10^18-1)

A357515 Smallest positive integer that doubles when the n rightmost digits are shifted to the left end.

Original entry on oeis.org

105263157894736842, 100502512562814070351758793969849246231155778894472361809045226130653266331658291457286432160804020

Offset: 1

Joseph C. Y. Wong, Oct 01 2022

a(n) with n>=3 is too large to be written in data.

The following is a method for finding a(n): Let n be the number of digits shifted, and let m be the smallest positive integer such that 10^m = 2 mod 2*10^n-1. We then look for the smallest positive b that is an n+d digit number and satisfies b = c(10^n-2)/(2*10^d-1), where c is a positive integer. Then a(n) = c(10^n-2)/(2*10^d-1)*10^n+c.

			a(1) = 105263157894736842 because shifting the 1 rightmost digit to the left end gives 210526315789473684 which is double a(1).

Cf. A146088

cp's OEIS Frontend

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

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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