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

1,2

Comments

This sequence appears to be infinite.

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

(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))
(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

Crossrefs

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

Cf. A029793, A257760, A257763, A257774.

Sequence in context: A330304 A272166 A291881 * A303738 A055578 A232733

Adjacent sequences: A257965 A257966 A257967 * A257969 A257970 A257971

Keyword

nonn,base

Author

Pieter Post, May 15 2015

Extensions

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

Status

approved