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A328849 Numbers in whose primorial base expansion only even digits appear.

%I A328849 #28 May 25 2023 07:27:21
%S A328849 0,4,12,16,24,28,60,64,72,76,84,88,120,124,132,136,144,148,180,184,
%T A328849 192,196,204,208,420,424,432,436,444,448,480,484,492,496,504,508,540,
%U A328849 544,552,556,564,568,600,604,612,616,624,628,840,844,852,856,864,868,900,904,912,916,924,928,960,964,972,976,984,988,1020,1024
%N A328849 Numbers in whose primorial base expansion only even digits appear.
%C A328849 Numbers for which the prime factor form (A276086) of their primorial base expansion is a square, A000290.
%H A328849 Antti Karttunen, <a href="/A328849/b328849.txt">Table of n, a(n) for n = 1..9072</a>
%H A328849 Antti Karttunen, <a href="/A328849/a328849.txt">Data supplement: n, a(n) computed for n = 1..90720</a>
%H A328849 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F A328849 a(n) = 2*A328770(n).
%F A328849 A000196(A276086(a(n))) = A276086(a(n)/2) = A328834(n).
%e A328849 144 is written as "4400" in primorial base (A049345), because 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus all the digits are even and 144 is included in this sequence.
%t A328849 With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[0, nmax, 2], AllTrue[prmBaseDigits[#], EvenQ] &]] (* _Amiram Eldar_, May 23 2023 *)
%o A328849 (PARI)
%o A328849 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A328849 isA328849(n) = issquare(A276086(n));
%Y A328849 Cf. A000290, A002110, A010052, A049345, A276086, A276156, A328770.
%Y A328849 Cf. A328834, A328850 (squares in this sequence).
%Y A328849 Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary), A351893 (factorial base).
%K A328849 nonn,base
%O A328849 1,2
%A A328849 _Antti Karttunen_, Oct 30 2019