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A256114 Numbers n such that digit_product(n^2) = (digit_product(n))^2 and n mod 10 > 0.

%I A256114 #28 May 22 2025 10:21:42
%S A256114 1,2,3,101,102,103,104,105,201,202,203,205,301,302,303,305,401,402,
%T A256114 403,405,501,502,503,504,505,506,507,508,509,601,602,603,605,609,661,
%U A256114 701,702,703,705,708,709,801,802,803,805,901,902,903,905,906,983
%N A256114 Numbers n such that digit_product(n^2) = (digit_product(n))^2 and n mod 10 > 0.
%C A256114 Contains i*10^d + j for i>=1, j mod 10 > 0, j < 10^d/(20*i+1). - _Robert Israel_, Jun 05 2015
%H A256114 Reiner Moewald, <a href="/A256114/b256114.txt">Table of n, a(n) for n = 1..15212</a>
%e A256114 digit_product(661^2) = digit_product(436921) = 1296 = 36^2 = (digit_product(661))^2.
%p A256114 pdigs:= n -> convert(convert(n,base,10),`*`):
%p A256114 select(t -> pdigs(t^2)=pdigs(t)^2, [seq(seq(10*k+j,j=1..9),k=0..1000)]); # _Robert Israel_, Jun 05 2015
%t A256114 pod[n_] := Times@@ IntegerDigits@ n; Select[ Range[10^4], Mod[#, 10] > 0 && pod[#]^2 == pod[#^2] &] (* _Giovanni Resta_, Jun 23 2015 *)
%o A256114 (Python)
%o A256114 def product_digits(n):
%o A256114    results = 1
%o A256114    while n > 0:
%o A256114       remainder = n % 10
%o A256114       results *= remainder
%o A256114       n = (n-remainder)/10
%o A256114    return results
%o A256114 pos = 0
%o A256114 for a in range(1,1000000):
%o A256114    if product_digits(a*a) == (product_digits(a))*(product_digits(a)) and (a%10 > 0):
%o A256114       pos += 1
%o A256114       print(pos, a)
%o A256114 (Magma) [t: j in [1..9], k in [0..100] | &*Intseq(t^2) eq &*Intseq(t)^2 where t is 10*k+j]; // _Bruno Berselli_, Jun 23 2015
%Y A256114 Cf. A007954, A256115.
%K A256114 nonn,base
%O A256114 1,2
%A A256114 _Reiner Moewald_, Mar 15 2015