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

2344, 172, 131, 174, 52, 77, 75, 83, 75, 81, 89, 95, 101, 104, 110, 133, 143, 127, 133, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 5

Author

Sascha Kurz, Oct 09 2001

Keywords

• base
• easy
• nonn

Comments

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.

Examples

			a(6) = 172 because 172 = [444]->[144]->[24]->[12]->[2] and no lesser n has persistence 4 in base 6.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 5..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 (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A003001, A031346, A064867, A064869, A064870, A064871, A064872.

Programs

Mathematica

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 *)

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);}
a(n) = {i=0; while (pers(i, n) != 4, i++); return (i);} \\ Michel Marcus, Jun 30 2013

Formula

a(n) = 5*n-floor(n/24) for n > 23.

From Chai Wah Wu, Mar 07 2025: (Start)

a(n) = a(n-1) + a(n-24) - a(n-25) for n > 48.

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)

Extensions

Example modified by Harvey P. Dale, Oct 19 2022