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

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

Offset: 1

Author

Seiichi Manyama, Aug 21 2017

Keywords

• nonn
• base

Comments

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

Examples

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

Links

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

Crossrefs

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.

Programs

Maple

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

Formula

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)