A159774 -id:A159774 - cp's OEIS frontend

A158877 Definition of a(n): in base-n arithmetic a(n) is the smallest positive integer that is doubled when its least significant digit is moved to become the most significant digit.

1012, 102, 13, 1031345242, 103524563142, 25, 10467842, 105263157894736842, 37, 10631694842

Offset: 3

Daniel Asimov (asimov(AT)msri.org), Mar 28 2009

The problem has no solution in base 2, so sequence begins with the base-3 solution. The idea was suggested by a NY Times article (Sunday Magazine of Mar 29, 2009) -- in which Freeman Dyson is said to have solved the base-10 question almost instantaneously when it was posed to him -- and by the ensuing math-fun discussion.

			For n = 5, the smallest positive integer whose base-5 representation doubles when the rightmost digit is moved to become the leftmost digit is 8 = 13_5; 31_5 = 16.
For n = 8, the smallest positive integer whose base-8 representation doubles when the rightmost digit is moved to become the leftmost digit is 21 = 25_8; 52_8 = 42. - _Robert Tanniru_, Aug 09 2022
For n = 13, the number can't be represented in this list as it would be 27A5 in base 13.

See A087502 (which is the main entry for this sequence) for these numbers written in base 10. Cf. A023094, A159774.

a(5) corrected by William A. Hoffman III (whoff(AT)robill.com), Apr 19 2009
a(8) corrected by Robert Tanniru, Aug 09 2022
a(11)-a(12) from Robert Tanniru, Aug 11 2022, using A087502

A147514 Least number m, written in base 10, 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

32, 18, 3472, 10993850, 2129428800, 546, 5064320, 105263157894736842, 380, 64609423538, 11424, 1673230, 58774271029236501660840264682112, 67650, 122181448512, 1666, 586081355679130611935159482937228562988190880, 210051282051282, 13571630704729343835960800

Offset: 3

Ray Chandler and R. J. Mathar, Apr 23 2009

Serves as an extension to A159774, which misses proper representation for solutions beyond base 12.
Algorithm: write m in base b with LSB d_0, k middle digits d_m, and MSB digit d_e as m=d_0+d_m*b+d_e*b^(k+1).
Demand m/2 = d_e+d_0*b_d_m*b^2 and 2*m=d_m+d_e*b^k+d_0*b^(k+1). Mix these to obtain m*(2b-1)=2*d_e*(b^(k+2)-1).
Loop over (outer loop) k=0,1,2... and (inner loop d_e=0.. b-1 to obtain integer m to be checked against the condition.

Cf. A159774.

A159863 a(3)(baseN) = least number m such that the quotient m/3 is obtained merely by shifting the leftmost digit (MSD) of m to the right end (LSD), and the multiple 3m by shifting the LSD of m to the MSD, in order of increasing number base, N.

Original entry on oeis.org

10113, 101343, 1020412245351433, 1023, 10262054413, 103, 1034482758620689655172413793, 2076

Offset: 4

William A. Hoffman III (whoff(AT)robill.com), Apr 24 2009

1001b2, 10b3 and 51b7 do not triple by specified shift and are not included although they allow MSD->LSD shift to make m/3. No 2-digit number qualifies because the shift is an exchange (e.g., 51b7/3=15b7). 2076b11 is the highest order number of this type to be represented without letter digits.

			2076b11/3=762b11 (2745/3=915) and 3*2076b11=6207b11 (3*2745=8235).

W. A. Hoffman III, Algorithm to compute terms.

Cf. A023060, A159774.

See A160116 for these numbers written in base 10.

Edited by Ray Chandler, May 02 2009

cp's OEIS Frontend

A158877 Definition of a(n): in base-n arithmetic a(n) is the smallest positive integer that is doubled when its least significant digit is moved to become the most significant digit.

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A147514 Least number m, written in base 10, 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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A159863 a(3)(baseN) = least number m such that the quotient m/3 is obtained merely by shifting the leftmost digit (MSD) of m to the right end (LSD), and the multiple 3m by shifting the LSD of m to the MSD, in order of increasing number base, N.

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