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A328850 Squares in whose primorial base expansion only even digits appear.

%I A328850 #18 Mar 06 2024 01:02:05
%S A328850 0,4,16,64,144,196,484,900,1024,1444,1764,2116,2304,4624,5184,5476,
%T A328850 6084,6724,13924,14400,14884,18496,19044,20164,23104,23716,24964,
%U A328850 28224,29584,61504,65536,66564,70756,73984,79524,80656,85264,88804,90000,121104,131044,135424,139876,186624,195364,204304,209764,242064,260100,264196
%N A328850 Squares in whose primorial base expansion only even digits appear.
%C A328850 Squares in A328849, squares such that also the prime factor form (A276086) of their primorial base expansion is a square,
%H A328850 Antti Karttunen, <a href="/A328850/b328850.txt">Table of n, a(n) for n = 1..390</a>
%H A328850 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F A328850 a(n) = A000290(A328838(n)).
%e A328850 12^2 = 144 is written as "4400" in primorial base (A049345), as 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus its prime code encoding, A276086(144) = prime(4)^4 * prime(3)^4 = 7^4 * 5^4 = 1500625 is also a square, and 144 is included in this sequence.
%t A328850 q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; AllTrue[s, EvenQ]]; Select[Range[0, 520]^2, q] (* _Amiram Eldar_, Mar 06 2024 *)
%o A328850 (PARI)
%o A328850 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A328850 isA328850(n) = (issquare(n) && issquare(A276086(n)));
%Y A328850 Cf. A328838 (gives the square roots).
%Y A328850 Intersection of A000290 and A328849.
%Y A328850 Cf. A002110, A010052, A049345, A276086.
%K A328850 nonn,base
%O A328850 1,2
%A A328850 _Antti Karttunen_, Oct 30 2019