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.

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

1,1

Comments

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.

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.

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

Links

Table of n, a(n) for n=1..31.

Example

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

Mathematica

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 *)

Prog

(Python)
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<j:
                ca=j
            ca=pod(ca, j)
            if c==d and ca==0:
                print (a, c)

Crossrefs

Cf. A003001, A014553, A135385.

Sequence in context: A262916 A262331 A128957 * A034277 A254702 A206615

Adjacent sequences: A258581 A258582 A258583 * A258585 A258586 A258587

Keyword

nonn,base

Author

Pieter Post, Nov 06 2015

Extensions

a(21)-a(31) from Jon E. Schoenfield, Nov 07 2015

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

Status

approved