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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.

%I A257968 #24 May 22 2025 10:21:43
%S A257968 1,2,38296,151373,398293,422558,733381,971973,2797318,3833215,6988327,
%T A257968 7271256,8174876,8732657,9872323,9981181,11617988,11798921,14791421,
%U A257968 15376465,15487926,15625186,16549885,18543639,21316582,21492828,22346329,22867986,23373644
%N 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.
%C A257968 This sequence appears to be infinite.
%H A257968 Giovanni Resta, <a href="/A257968/b257968.txt">Table of n, a(n) for n = 1..10000</a>
%F A257968 pod(n^3)/pod(n^2)=pod(n^2)/pod(n), where pod(n) = A007954(n).
%e A257968 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.
%t A257968 pod[n_]:=Times@@IntegerDigits@n; Select[Range[10^8], pod[#^3] pod[#] == pod[#^2]^2 >0 &] (* _Vincenzo Librandi_, May 16 2015 *)
%o A257968 (Python)
%o A257968 def pod(n):
%o A257968     kk = 1
%o A257968     while n > 0:
%o A257968         kk= kk*(n%10)
%o A257968         n =int(n//10)
%o A257968     return kk
%o A257968 for i in range (1,10**7):
%o A257968     if pod(i**3)*pod(i)==pod(i**2)**2 and pod(i**2)!=0:
%o A257968         print (i, pod(i),pod(i**2),pod(i**3),pod(i**2)//pod(i))
%o A257968 (PARI) pod(n) = my(d = digits(n)); prod(k=1, #d, d[k]);
%o A257968 isok(n) = (pd = pod(n)) && (pdd = pod(n^2)) && (pdd/pd == pod(n^3)/pdd); \\ _Michel Marcus_, May 30 2015
%Y A257968 Cf. A052382 (zeroless numbers), A007954 (product of digits).
%Y A257968 Cf. A029793, A257760, A257763, A257774.
%K A257968 nonn,base
%O A257968 1,2
%A A257968 _Pieter Post_, May 15 2015
%E A257968 a(17)-a(29) from _Giovanni Resta_, May 15 2015