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(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).
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
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(*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}]
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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
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
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).
See
A147514 for these numbers written in base 10.
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
A094224
Number of digits in the least n-transposable number.
Original entry on oeis.org
18, 28, 6, 6, 58, 22, 13, 44
Offset: 2
We have the a(4)=6-digit 4-transposable number 410256=4*102564, and the a(5)=6-digit 5-transposable number 714285=5*142857.
- P. Tougne, "Jeux Mathématiques", Prob. 8, pp. 104 and 107 Aug. 1982 issue of Pour La Science (French edition of 'Scientific American'), Paris.
Cf.
A092697 (value of the least n-transposable number).
A317526
Smallest "Shift and erase" numbers beginning by (9 + n). See the Comments section.
Original entry on oeis.org
1010, 11100917431192660550458715596330275229357798165137614678899082568807339449541284403669724770642201834862385321, 12100840336134453781512605042016806722689075630252, 13100775193798449612403, 141007194244604316546762589928057553956834532374
Offset: 1
To divide a(1) = 1010 by 10, just shift the first digit 1 to the end and erase 0 (which gives 101 -- and, indeed, 101*10 = 1010).
To divide a(4) = 13100775193798449612403 by 13, just shift the first digit 1 to the end and erase 3 (which gives 1007751937984496124031 -- and, indeed, 1007751937984496124031*13 = 13100775193798449612403).
Cf.
A092697 (least n-parasitic numbers).
A353054
Numbers k such that placing the last digit first gives 2k+1.
Original entry on oeis.org
1052, 26315, 15789473, 3157894736, 421052631578, 2105263157894, 36842105263157, 1052631578947368421052, 26315789473684210526315, 15789473684210526315789473, 3157894736842105263157894736, 421052631578947368421052631578, 2105263157894736842105263157894, 36842105263157894736842105263157
Offset: 1
2*1052 + 1 = 2105. Thus, 1052 is in this sequence.
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Select[Range[100000000], FromDigits[Prepend[Drop[IntegerDigits[#], -1], Last[IntegerDigits[#]]]] == 2 # + 1 &]
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f(n) = if (n < 10, n, my(d=digits(n)); fromdigits(concat(d[#d], Vec(d, #d-1))));
isok(m) = f(m) == 2*m+1; \\ Michel Marcus, Apr 21 2022
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from itertools import count, islice
def A353054_gen(): # generator of terms
for l in count(1):
a, b = 10**l-2, 10**(l-1)-2
for m in range(1,10):
q, r = divmod(m*a-1,19)
if r == 0 and b <= q - 2 <= a:
yield 10*q+m
A353054_list = list(islice(A353054_gen(),20)) # Chai Wah Wu, Apr 23 2022
Comments