A046148
Number of n-digit numbers with maximal multiplicative persistence A014553.
Original entry on oeis.org
10, 1, 9, 12, 20, 2430, 5229, 7448, 282852, 88200, 8015040, 200676960, 2701775518, 24655323238, 15765750, 1715313600, 59049874884, 1112489914536, 14162129381400, 135917876094000, 1050596838951660, 6832549561749912, 38554260751029408, 193081920969057120
Offset: 1
-
mper[n_] := Block[{k=0, m=n}, While[m>9, k++; m = Times @@ IntegerDigits@ m]; k]; mxper = {1, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10}; multi[w_] := Total[w]!/Times @@ (w!); a[1]=10; a[n_] := Sum[ Total[ If[ mxper[[n]] == 1 + mper[Times @@ (Range[9]^#)], multi[#], 0] & /@ Permutations[p]], {p, IntegerPartitions[n, {9}, Range[0, n]]}]; Array[a, 12] (* Giovanni Resta, Sep 01 2018 *)
A046150
Largest n-digit number with maximal multiplicative persistence A014553.
Original entry on oeis.org
9, 77, 976, 8876, 98886, 997762, 9999996, 99988862, 999888621, 9998888773, 99988887731, 999888877311, 9998888773111, 99988887731111, 998888887777772, 9988888877777721, 99999999998777772, 999999999987777721, 9999999999877777211, 99999999998777772111
Offset: 1
A046149
Smallest n-digit number with maximal multiplicative persistence A014553.
Original entry on oeis.org
0, 77, 679, 6788, 68889, 168889, 2677889, 26888999, 126888999, 3778888999, 13778888999, 113778888999, 1113778888999, 11113778888999, 277777788888899, 1277777788888899, 11277777788888899, 111277777788888899
Offset: 1
A035927
One less than number of n-multisets chosen from a 10-set.
Original entry on oeis.org
0, 9, 54, 219, 714, 2001, 5004, 11439, 24309, 48619, 92377, 167959, 293929, 497419, 817189, 1307503, 2042974, 3124549, 4686824, 6906899, 10015004, 14307149, 20160074, 28048799, 38567099, 52451255, 70607459, 94143279
Offset: 0
- Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 56, Cambridge, MA: Mass. Institute of Technology Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972, Item 56.
- Eric Weisstein's World of Mathematics, Multiplicative Persistence.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
-
binomial(10+n-1,n)-1;
-
Table[Binomial[9 + n, n] - 1, {n, 0, 27}] (* Michael De Vlieger, Jul 14 2015 *)
CoefficientList[Series[1/(1-x)^10-1/(1-x),{x,0,30}],x] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,9,54,219,714,2001,5004,11439,24309,48619},30] (* Harvey P. Dale, Jul 11 2023 *)
-
a(n)=if(n<0,0,binomial(n+9,9)-1)
A258584
Numbers n such that n = Sum_{j>=1} c(j) where c(0) = n, c(j) = floor(c(j-1)/10^k)*(c(j-1) mod 10^k) for j>0, and k is half the number of digits of n, rounded up if the number of digits of n is odd.
Original entry on oeis.org
86, 860, 1975, 2160, 3575, 19750, 21600, 35750, 43614, 51884, 65625, 479900, 868688, 967750, 1435575, 1548384, 1696875, 4799000, 8686880, 9677500, 28874200, 34095100, 38748800, 39214560, 47613625, 53415625, 148385715, 156293216, 288742000, 340951000, 387488000
Offset: 1
86 is in the sequence because 8*6 = 48, 4*8 = 32 and 3*2 = 6. And 86 = 48 + 32 + 6.
-
fQ[n_] := Block[{i = Ceiling[IntegerLength[n]/2], g}, g[x_] := If[IntegerLength@ x <= i, x, Times @@ (FromDigits /@ {If[IntegerLength@ x - i == 0, Nothing, Take[IntegerDigits@ x, IntegerLength@ x - i]], Take[IntegerDigits@ x, -i]})]; Total@ Rest@ Most@ FixedPointList[g, n] == n]; Select[Range@ 500000, fQ] (* Michael De Vlieger, Nov 06 2015 *)
-
def pod(n, m):
kk = 1
while n > 0:
kk= kk*(n%m)
n =int(n//m)
return kk
for b in range(0, 6):
dd, bb=0, (b-1)//2+2
j=10**bb
for c in range (10*j, 100*j):
d, a, ca=0, 0, pod(c, j)
while ca>0:
d, a=d+ca, a+1
if ca
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