A014120 Smallest number of persistence n over product-of-nonzero-digits function.

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 267777777889999, 77777777788888888888899999, 37777777777777777777777777778888889999999999999999999

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• base
• nice

Comments

Comments from Marc Lapierre, Sep 09 2020, updated Feb 19 2025: (Start)

Let d(b) denote a string of b copies of the digit d. For example, 3(5) would represent 33333.

Here are recent discoveries and the initials of the discoverers:

14 2(1)6(1)7(1)8(99)9(10) by WW

15 6(1)7(157)8(46)9(25) by WW

16 3(1)7(54)8(82)9(353) by WW

17 3(1)7(27)8(622)9(399) by WW

18 3(1)7(140)8(258)9(1946) by WW

19 2(1)7(122)8(498)9(4297) by ML

20 2(1)7(723)8(211)9(9825) by ML

Conjectured terms follow:

21 2(1)6(1)7(822)8(29)9(22601) by CC

22 2(1)7(325)8(1678)9(49461) by CC

23 2(1)7(1199)8(662)9(110077) by CC

24 7(1335)8(1609)9(241857) by CC

25 4(1)7(155)8(1135)9(540199) by CC

26 6(1)7(124)8(762)9(1192494) by CC

27 4(1)7(94)8(583)9(2616367) by CC

28 3(1)5(6)9(5788986) by CC

29 7(20)8(24)9(12878515) by CC

30 8(67)9(29136478) by CC

31 7(9)8(17)9(64768493) by CC

32 6(1)7(8)8(5)9(144417969) by CC

33 6(1)7(2)8(8)9(324379548) by CC

34 2(1)6(1)7(1)8(31)9(728759871) by CC

35 2(1)6(1)8(555702)9(1553958443) by CC

36 3(1)8(10311016)9(3652133953) by CC

37 2(1)6(1)8(1482224)9(9136624827) by CC

38 3(1)8(1469099)9(19262606200) by CC

39 8(30408394)9(42905304588) by CC

WW=Wilfred Whiteside & Phil Carmody (see link)

ML=Marc Lapierre

CC=Christophe Clavier

It seems a safe conjecture that this sequence is infinite.

(End)

Links

• Marc Lapierre, Table of n, a(n) for n = 0..16
• Marc Lapierre, Multiplicative persistence computation
• N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
• Wilfred Whiteside & Phil Carmody, Puzzle 341. Multiplicative persistence, Erdos style, Problems & Puzzles: Puzzles.

Crossrefs

Cf. A003001.

Extensions

Edited by N. J. A. Sloane, Sep 11 2020 following suggestions from Marc Lapierre, Sep 09 2020