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

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

Offset: 0

Author

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

Keywords

• nonn
• base

Comments

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

Links

• Wikipedia, Parasitic number.

Crossrefs

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.

Programs

PARI

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

Formula

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

Extensions

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