Multiplicative persistence is a complex term but a simple concept. Take a number, multiply its digits, repeat. Sooner or later the result is a single digit:

25 → 2 x 5 = 10 → 1 x 0 = 0 (mp=2)

39 → 3 x 9 = 27 → 2 x 7 = 14 → 1 x 4 = 4 (mp=3)

So 25 has a multiplicative persistence of 2 and 39 a multiplicative persistence of 3. Each is the smallest number with that m.p. in base-10. Further records are set by these numbers:

77 → 49 → 36 → 18 → 8 (mp=4)

679 → 378 → 168 → 48 → 32 → 6 (mp=5)

6788 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=6)

68889 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=7)

2677889 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=8)

26888999 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=9)

3778888999 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=10)

Now here’s base-9:

25_{[b=9]} → 11 → 1 (mp=2)

38_{[b=9]} → 26 → 13 → 3 (mp=3)

57_{[b=9]} → 38 → 26 → 13 → 3 (mp=4)

477_{[b=9]} → 237 → 46 → 26 → 13 → 3 (mp=5)

45788_{[b=9]} → 13255 → 176 → 46 → 26 → 13 → 3 (mp=6)

2577777_{[b=9]} → 275484 → 13255 → 176 → 46 → 26 → 13 → 3 (mp=7)

And base-11:

26_{[b=11]} → 11 → 1 (mp=2)

3A_{[b=11]} → 28 → 15 → 5 (mp=3)

69_{[b=11]} → 4A → 37 → 1A → A (=10_{b=10}) (mp=4)

269_{[b=11]} → 99 → 74 → 26 → 11 → 1 (mp=5)

3579_{[b=11]} → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=6)

26778_{[b=11]} → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=7)

47788A_{[b=11]} → 86277 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=8)

67899AAA_{[b=11]} → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=9)

77777889999_{[b=11]} → 2AA174996A → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=10)

I was also interested in the narcissism of multiplicative persistence. That is, are any numbers equal to the sum of the numbers created while calculating their multiplicative persistence? Yes:

86 = (8 x 6 = 48) + (4 x 8 = 32) + (3 x 2 = 6)

I haven’t found any more in base-10 (apart from the trivial 0 to 9) and can’t prove that this is the only one. Base-9 offers this:

78_{[b=9]} = 62 + 13 + 3

I can’t find any at all in base-11, but here are base-12 and base-27:

57_{[b=12]} = 2B + 1A + A

A8_{[b=12]} = 68 + 40 + 0

4[23]_{[b=27]} = 3B + 16 + 6

7[24]_{[b=27]} = 66 + 19 + 9

A[18]_{[b=27]} = 6[18] + 40 + 0

[26][24]_{[b=27]} = [23]3 + 2F + 13 + 3

[26][23][26]_{[b=27]} = [21]8[23] + 583 + 4C + 1[21] + [21]

But the richest base I’ve found so far is base-108, with fourteen narcissistic multiplicative-persistence sums:

4[92]_{[b=108]} = 3[44] + 1[24] + [24]

5[63]_{[b=108]} = 2[99] + 1[90] + [90]

7[96]_{[b=108]} = 6[24] + 1[36] + [36]

A[72]_{[b=108]} = 6[72] + 40 + 0

[19][81]_{[b=108]} = E[27] + 3[54] + 1[54] + [54]

[26][96]_{[b=108]} = [23]C + 2[60] + 1C + C

[35][81]_{[b=108]} = [26][27] + 6[54] + 30 + 0

[37][55]_{[b=108]} = [18][91] + F[18] + 2[54] + 10 + 0

[73][60]_{[b=108]} = [40][60] + [22][24] + 4[96] + 3[60] + 1[72] + [72]

[107][66]_{[b=108]} = [65][42] + [25][30] + 6[102] + 5[72] + 3[36] + 10 + 0

[71][84]_{[b=108]} = [55][24] + C[24] + 2[72] + 1[36] + [36]

[107][99]_{[b=108]} = [98]9 + 8[18] + 1[36] + [36]

5[92][96]_{[b=108]} = 3[84][96] + 280 + 0

8[107][100]_{[b=108]} = 7[36][64] + 1[41][36] + D[72] + 8[72] + 5[36] + 1[72] + [72]

Update (10/ii/14): The best now is base-180 with eighteen multiplicative-persistence sums.

5[105]_{[b=180]} = 2[165] + 1[150] + [150]

7[118]_{[b=180]} = 4[106] + 2[64] + [128]

7[160]_{[b=180]} = 6[40] + 1[60] + [60]

8[108]_{[b=180]} = 4[144] + 3[36] + [108]

A[120]_{[b=180]} = 6[120] + 40 + 0 (s=5)

[19][135]_{[b=180]} = E[45] + 3[90] + 1[90] + [90]

[21][108]_{[b=180]} = C[108] + 7[36] + 1[72] + [72]

[26][160]_{[b=180]} = [23][20] + 2[100] + 1[20] + [20]

[31][98]_{[b=180]} = [16][158] + E8 + [112]

[35][135]_{[b=180]} = [26][45] + 6[90] + 30 + 0 (s=10)

[44][96]_{[b=180]} = [23][84] + A[132] + 7[60] + 2[60] + [120]

[71][140]_{[b=180]} = [55][40] + C[40] + 2[120] + 1[60] + [60]

[73][100]_{[b=180]} = [40][100] + [22][40] + 4[160] + 3[100] + 1[120] + [120]

[107][110]_{[b=180]} = [65][70] + [25][50] + 6[170] + 5[120] + 3[60] + 10 + 0

[107][165]_{[b=180]} = [98]F + 8[30] + 1[60] + [60] (s=15)

[172][132]_{[b=180]} = [126][24] + [16][144] + C[144] + 9[108] + 5[72] + 20 + 0

5[173][145]_{[b=180]} = 3[156][145] + 2[17]0 + 0

E[170][120]_{[b=180]} = 8[146][120] + 4[58][120] + [154][120] + [102][120] + [68]0 + 0