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

%I A291215 #29 May 04 2025 03:25:13
%S A291215 1014492753623188405797,1159420289855072463768,1304347826086956521739,
%T A291215 10144927536231884057971014492753623188405797,
%U A291215 11594202898550724637681159420289855072463768,13043478260869565217391304347826086956521739,101449275362318840579710144927536231884057971014492753623188405797
%N A291215 Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 7.
%C A291215 With x = (10^21 - 7)/69 = 14492753623188405797, we have
%C A291215   a(1) = 7*x*10 + 7, a(2) = 8*x*10 + 8, a(3) = 9*x*10 + 9.
%C A291215 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
%H A291215 Robert Israel, <a href="/A291215/b291215.txt">Table of n, a(n) for n = 1..135</a>
%H A291215 Wikipedia, <a href="https://en.wikipedia.org/wiki/Parasitic_number">Parasitic number</a>.
%F A291215 From _Robert Israel_, Aug 22 2017: (Start)
%F A291215 a(3k-2) = 7(10^(22k)-1)/69.
%F A291215 a(3k-1) = 8(10^(22k)-1)/69.
%F A291215 a(3k) = 9(10^(22k)-1)/69.
%F A291215 a(n+6) = (10^22+1) a(n+3) - 10^22 a(n).
%F A291215 G.f.: x*(1304347826086956521739*x^2 + 1159420289855072463768*x + 1014492753623188405797)/(10^22*x^6 - (10^22+1)*x^3 + 1). (End)
%e A291215 b = 101449275362318840579.
%e A291215 a(1) = b*10 + 7,
%e A291215 7*a(1) = 7101449275362318840579 = 7*10^21 + b.
%p A291215 seq(seq(y*((10^(22*k)-1)/69),y=7..9),k=1..6); # _Robert Israel_, Aug 22 2017
%Y A291215 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).
%Y A291215 All these are subsequences of A034089 (except for an initial 0 in some of them).
%Y A291215 Cf. A092697, A097717.
%K A291215 nonn,base
%O A291215 1,1
%A A291215 _Seiichi Manyama_, Aug 21 2017