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A064867 The minimal number which has multiplicative persistence 3 in base n.

%I A064867 #21 Mar 07 2025 16:12:41
%S A064867 26,63,68,23,27,31,35,39,43,46,50,54,58,62,66,69,73,77,81,85,89,92,96,
%T A064867 100,104,108,112,115,119,123,127,131,135,138,142,146,150,154,158,161,
%U A064867 165,169,173,177,181,184,188,192
%N A064867 The minimal number which has multiplicative persistence 3 in base n.
%C A064867 The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.
%H A064867 Michael De Vlieger, <a href="/A064867/b064867.txt">Table of n, a(n) for n = 3..10000</a>
%H A064867 M. R. Diamond and D. D. Reidpath, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/persistence/PERSIST.PDF">A counterexample to a conjecture of Sloane and Erdos</a>, J. Recreational Math., 1998 29(2), 89-92.
%H A064867 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/persistence/persistence.html">Persistence in different bases</a>
%H A064867 T. Lamont-Smith, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Lamont/lamont5.html">Multiplicative Persistence and Absolute Multiplicative Persistence</a>, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
%H A064867 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_022.htm">Minimal prime with persistence p</a>
%H A064867 N. J. A. Sloane, <a href="http://neilsloane.com/doc/persistence.html">The persistence of a number</a>, J. Recreational Math., 6 (1973), 97-98.
%H A064867 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%H A064867 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A064867 a(n) = 4*n-floor(n/6) for n > 5.
%F A064867 From _Chai Wah Wu_, Mar 07 2025: (Start)
%F A064867 a(n) = a(n-1) + a(n-6) - a(n-7) for n > 12.
%F A064867 G.f.: x^3*(48*x^9 - x^8 - 33*x^7 - 22*x^6 + 4*x^5 + 4*x^4 - 45*x^3 + 5*x^2 + 37*x + 26)/(x^7 - x^6 - x + 1). (End)
%e A064867 a(3) = 26 because 26 = [222]->[22]->[11]->[1] and no fewer n has persistence 3 in base 3.
%t A064867 With[{m = 3}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k, 100] != m + 2, k++]; k], {n, 3, 5}]]~Join~Array[4 # - Floor[#/6] &, 45, 6] (* _Michael De Vlieger_, Aug 30 2021 *)
%Y A064867 Cf. A003001, A031346, A064868, A064869, A064870, A064871, A064872.
%K A064867 base,easy,nonn
%O A064867 3,1
%A A064867 _Sascha Kurz_, Oct 08 2001