A258584 - OEIS

%I #33 Dec 13 2023 08:50:15

%S 86,860,1975,2160,3575,19750,21600,35750,43614,51884,65625,479900,

%T 868688,967750,1435575,1548384,1696875,4799000,8686880,9677500,

%U 28874200,34095100,38748800,39214560,47613625,53415625,148385715,156293216,288742000,340951000,387488000

%N 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.

%C If n is an odd-digit decimal number, the first half is one digit smaller than the second half. For example, 43614 is in the sequence, because 43*614 = 26402, 26*402 = 10452, 10*452 = 4520, 4*520 = 2080, 2*80 = 160. Here the iteration stops because 160 has three digits, so the first half of the next multiplication is zero. 43614 = 26402 + 10452 + 4520 + 2080 + 160.

%C If n is an even-digit decimal number, the first half and the second half have the same length. For example, 868688 is in the sequence because 868*688 = 597184, 597*184 = 109848, 109*848 = 92432, 92*432 = 39744, 39*744 = 29016, 29*16 = 464, and here the iteration stops. 868688 = 597184 + 109848 + 92432 + 39744 + 29016 + 464.

%C If n is in the sequence and has an even number of digits, then 10*n is also in the sequence. - _Jon E. Schoenfield_, Nov 07 2015

%e 86 is in the sequence because 8*6 = 48, 4*8 = 32 and 3*2 = 6. And 86 = 48 + 32 + 6.

%t 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 *)

%o (Python)

%o def pod(n, m):

%o     kk = 1

%o     while n > 0:

%o         kk= kk*(n%m)

%o         n =int(n//m)

%o     return kk

%o for b in range(0, 6):

%o     dd, bb=0, (b-1)//2+2

%o     j=10**bb

%o     for c in range (10*j, 100*j):

%o         d, a, ca=0, 0, pod(c, j)

%o         while ca>0:

%o             d, a=d+ca, a+1

%o             if ca<j:

%o                 ca=j

%o             ca=pod(ca, j)

%o             if c==d and ca==0:

%o                 print (a, c)

%Y Cf. A003001, A014553, A135385.

%K nonn,base

%O 1,1

%A _Pieter Post_, Nov 06 2015

%E a(21)-a(31) from _Jon E. Schoenfield_, Nov 07 2015

%E Obsolete b-file deleted by _N. J. A. Sloane_, Jan 05 2019