A064867 The minimal number which has multiplicative persistence 3 in base n.
26, 63, 68, 23, 27, 31, 35, 39, 43, 46, 50, 54, 58, 62, 66, 69, 73, 77, 81, 85, 89, 92, 96, 100, 104, 108, 112, 115, 119, 123, 127, 131, 135, 138, 142, 146, 150, 154, 158, 161, 165, 169, 173, 177, 181, 184, 188, 192
Offset: 3
Examples
a(3) = 26 because 26 = [222]->[22]->[11]->[1] and no fewer n has persistence 3 in base 3.
Links
- Michael De Vlieger, Table of n, a(n) for n = 3..10000
- M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
- Sascha Kurz, Persistence in different bases
- T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
- C. Rivera, Minimal prime with persistence p
- N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
- Eric Weisstein's World of Mathematics, Multiplicative Persistence
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).
Programs
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Mathematica
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 *)
Formula
a(n) = 4*n-floor(n/6) for n > 5.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n > 12.
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)
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