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

0, 4, 16, 64, 144, 196, 484, 900, 1024, 1444, 1764, 2116, 2304, 4624, 5184, 5476, 6084, 6724, 13924, 14400, 14884, 18496, 19044, 20164, 23104, 23716, 24964, 28224, 29584, 61504, 65536, 66564, 70756, 73984, 79524, 80656, 85264, 88804, 90000, 121104, 131044, 135424, 139876, 186624, 195364, 204304, 209764, 242064, 260100, 264196

Offset: 1

Author

Antti Karttunen, Oct 30 2019

Keywords

• nonn
• base

Comments

Squares in A328849, squares such that also the prime factor form (A276086) of their primorial base expansion is a square,

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..390
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A328838 (gives the square roots).

Intersection of A000290 and A328849.

Cf. A002110, A010052, A049345, A276086.

Programs

Mathematica

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 *)

PARI

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328850(n) = (issquare(n) && issquare(A276086(n)));

Formula

a(n) = A000290(A328838(n)).