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A064868 The minimal number which has multiplicative persistence 4 in base n.

%I A064868 #26 Mar 07 2025 16:12:37
%S A064868 2344,172,131,174,52,77,75,83,75,81,89,95,101,104,110,133,143,127,133,
%T A064868 119,124,129,134,139,144,149,154,159,164,169,174,179,184,189,194,199,
%U A064868 204,209,214,219,224,229,234,238,243,248,253,258,263,268,273,278,283
%N A064868 The minimal number which has multiplicative persistence 4 in base n.
%C A064868 The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) do not seem to exist.
%H A064868 Michael De Vlieger, <a href="/A064868/b064868.txt">Table of n, a(n) for n = 5..10000</a>
%H A064868 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 A064868 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/persistence/persistence.html">Persistence in different bases</a>
%H A064868 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 A064868 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_022.htm">Minimal prime with persistence p</a>
%H A064868 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 A064868 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%H A064868 <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A064868 a(n) = 5*n-floor(n/24) for n > 23.
%F A064868 From _Chai Wah Wu_, Mar 07 2025: (Start)
%F A064868 a(n) = a(n-1) + a(n-24) - a(n-25) for n > 48.
%F A064868 G.f.: x^5*(18*x^43 - x^42 + 21*x^41 - 5*x^40 - 18*x^39 - x^38 + 2*x^37 - x^36 - x^35 - 3*x^34 - x^33 + 13*x^32 - 3*x^31 + 7*x^30 - 20*x^29 + 127*x^28 - 38*x^27 + 46*x^26 + 2177*x^25 - 2339*x^24 + 5*x^23 + 5*x^22 + 5*x^21 + 5*x^20 - 14*x^19 + 6*x^18 - 16*x^17 + 10*x^16 + 23*x^15 + 6*x^14 + 3*x^13 + 6*x^12 + 6*x^11 + 8*x^10 + 6*x^9 - 8*x^8 + 8*x^7 - 2*x^6 + 25*x^5 - 122*x^4 + 43*x^3 - 41*x^2 - 2172*x + 2344)/(x^25 - x^24 - x + 1). (End)
%e A064868 a(6) = 172 because 172 = [444]->[144]->[24]->[12]->[2] and no lesser n has persistence 4 in base 6.
%t A064868 With[{m = 4, r = 24}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k] != m + 2, k++]; k], {n, m + 1, r}]~Join~Array[(m + 1) # - Floor[#/r] &, 34, r + 1]] (* _Michael De Vlieger_, Aug 30 2021 *)
%o A064868 (PARI) pers(nn, b) = {ok = 0; p = 0; until (ok, d = digits(nn, b); if (#d == 1, ok = 1, p++); nn = prod(k=1, #d, d[k]); if (nn == 0, ok = 1);); return (p);}
%o A064868 a(n) = {i=0; while (pers(i, n) != 4, i++); return (i);} \\ _Michel Marcus_, Jun 30 2013
%Y A064868 Cf. A003001, A031346, A064867, A064869, A064870, A064871, A064872.
%K A064868 base,easy,nonn
%O A064868 5,1
%A A064868 _Sascha Kurz_, Oct 09 2001
%E A064868 Example modified by _Harvey P. Dale_, Oct 19 2022