A328850 -id:A328850 - cp's OEIS frontend

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

0, 4, 12, 16, 24, 28, 60, 64, 72, 76, 84, 88, 120, 124, 132, 136, 144, 148, 180, 184, 192, 196, 204, 208, 420, 424, 432, 436, 444, 448, 480, 484, 492, 496, 504, 508, 540, 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

Offset: 1

Antti Karttunen, Oct 30 2019

Numbers for which the prime factor form (A276086) of their primorial base expansion is a square, A000290.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..9072
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..90720
Index entries for sequences related to primorial base.

Cf. A000290, A002110, A010052, A049345, A276086, A276156, A328770.
Cf. A328834, A328850 (squares in this sequence).
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary), A351893 (factorial base).

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 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328849(n) = issquare(A276086(n));

a(n) = 2*A328770(n).

A000196(A276086(a(n))) = A276086(a(n)/2) = A328834(n).

A328838 Numbers such that in the primorial base expansion of their squares only even digits appear.

Original entry on oeis.org

0, 2, 4, 8, 12, 14, 22, 30, 32, 38, 42, 46, 48, 68, 72, 74, 78, 82, 118, 120, 122, 136, 138, 142, 152, 154, 158, 168, 172, 248, 256, 258, 266, 272, 282, 284, 292, 298, 300, 348, 362, 368, 374, 432, 442, 452, 458, 492, 510, 514, 548, 558, 562, 574, 608, 616, 652, 660, 698, 704, 708, 1018, 1020, 1042, 1054, 1080, 1082, 1096, 1124

Offset: 1

Antti Karttunen, Oct 30 2019

			For n = 4, its square 16 is written as "220" in primorial base (A049345), as 2*A002110(2) + 2*A002110(1) + 0*A002110(0) = 2*6 + 2*2 = 16, thus 4 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..19309
Index entries for sequences related to primorial base.

Cf. A000196, A002110, A010052, A049345, A276086, A328849, A328850.

q[n_] := Module[{k = n^2, 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, 1200], q] (* Amiram Eldar, Mar 06 2024 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328838(n) = (issquare(A276086(n*n)));

a(n) = A000196(A328850(n)).

cp's OEIS Frontend

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

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