A257968 Zeroless numbers n such that the product of digits of n, the product of digits of n^2 and the product of digits of n^3 form a geometric progression.

1, 2, 38296, 151373, 398293, 422558, 733381, 971973, 2797318, 3833215, 6988327, 7271256, 8174876, 8732657, 9872323, 9981181, 11617988, 11798921, 14791421, 15376465, 15487926, 15625186, 16549885, 18543639, 21316582, 21492828, 22346329, 22867986, 23373644

Offset: 1

Author

Pieter Post, May 15 2015

Keywords

• nonn
• base

Comments

This sequence appears to be infinite.

Examples

			38296 is in the sequence because the pod equals 2592 (=3*8*2*9*6), pod(38296^2) is 622080, pod(38296^3) is 149299200. 2592*240 = 622080 => 622080*240 = 149299200.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052382 (zeroless numbers), A007954 (product of digits).

Cf. A029793, A257760, A257763, A257774.

Programs

Mathematica

pod[n_]:=Times@@IntegerDigits@n; Select[Range[10^8], pod[#^3] pod[#] == pod[#^2]^2 >0 &] (* Vincenzo Librandi, May 16 2015 *)

PARI

pod(n) = my(d = digits(n)); prod(k=1, #d, d[k]);
isok(n) = (pd = pod(n)) && (pdd = pod(n^2)) && (pdd/pd == pod(n^3)/pdd); \\ Michel Marcus, May 30 2015

Python

def pod(n):
    kk = 1
    while n > 0:
        kk= kk*(n%10)
        n =int(n//10)
    return kk
for i in range (1,10**7):
    if pod(i**3)*pod(i)==pod(i**2)**2 and pod(i**2)!=0:
        print (i, pod(i),pod(i**2),pod(i**3),pod(i**2)//pod(i))

Formula

pod(n^3)/pod(n^2)=pod(n^2)/pod(n), where pod(n) = A007954(n).

Extensions

a(17)-a(29) from Giovanni Resta, May 15 2015